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User:Jon Awbrey/SCRATCHPAD (view source)

Revision as of 15:04, 2 February 2009

35 bytes added ,  15:04, 2 February 2009
→‎Rule 10
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===Rule 10===
 
===Rule 10===
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<pre>
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Rule 10
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If X, Y c U,
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then the following are equivalent:
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R10a. X = Y. :D2a
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::
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R10b. u C X  <=>  u C Y, for all u C U. :D2b
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:R8a
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::
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R10c. [u C X] = [u C Y]. :R8b
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::
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R10d. For all u C U,
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[u C X](u) = [u C Y](u). :R8c
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::
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R10e. ConjUu ( [u C X](u)  =  [u C Y](u) ). :R8d
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::
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R10f. ConjUu ( [u C X](u) <=> [u C Y](u) ). :R8e
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::
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R10g. ConjUu (( [u C X](u) , [u C Y](u) )). :R8f
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::
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R10h. ConjUu (( [u C X] , [u C Y] ))$(u). :R8g
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</pre>
      
<br>
 
<br>
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| width="18%" style="border-top:1px solid black" | <math>\operatorname{R10a.}</math>
 
| width="18%" style="border-top:1px solid black" | <math>\operatorname{R10a.}</math>
 
| width="60%" style="border-top:1px solid black" | <math>P ~=~ Q</math>
 
| width="60%" style="border-top:1px solid black" | <math>P ~=~ Q</math>
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| width="20%" style="border-top:1px solid black; border-left:1px solid black; text-align:center" | <math>\operatorname{R10a~:~R5a}</math>
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| width="20%" style="border-top:1px solid black; border-left:1px solid black; text-align:center" | <math>\operatorname{R10a~:~D2a}</math>
 
|- style="height:20px"
 
|- style="height:20px"
 
| &nbsp;
 
| &nbsp;
Line 2,969: Line 2,943:  
| &nbsp;
 
| &nbsp;
 
| <math>\operatorname{R10b.}</math>
 
| <math>\operatorname{R10b.}</math>
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| <math>\upharpoonleft P \upharpoonright ~=~ \upharpoonleft Q \upharpoonright</math>
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| <math>\overset{X}{\underset{x}{\forall}}~ (x \in P ~\Leftrightarrow~ x \in Q)</math>
 
| style="border-left:1px solid black; text-align:center" |
 
| style="border-left:1px solid black; text-align:center" |
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<p><math>\operatorname{R10b~:~R5e}</math></p>
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<p><math>\operatorname{R10b~:~D2b}</math></p>
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<p><math>\operatorname{R10b~:~R7a}</math></p>
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<p><math>\operatorname{R10b~:~R8a}</math></p>
 
|- style="height:20px"
 
|- style="height:20px"
 
| &nbsp;
 
| &nbsp;
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| &nbsp;
 
| &nbsp;
 
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
 
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:60px"
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|- style="height:40px"
 
| &nbsp;
 
| &nbsp;
 
| <math>\operatorname{R10c.}</math>
 
| <math>\operatorname{R10c.}</math>
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| <math>\overset{X}{\underset{x}{\forall}}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))</math>
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| <math>\downharpoonleft x \in P \downharpoonright ~=~ \downharpoonleft x \in Q \downharpoonright</math>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10c~:~R7b}</math>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10c~:~R8b}</math>
 
|- style="height:20px"
 
|- style="height:20px"
 
| &nbsp;
 
| &nbsp;
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| &nbsp;
 
| &nbsp;
 
| <math>\operatorname{R10d.}</math>
 
| <math>\operatorname{R10d.}</math>
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| <math>\operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))</math>
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| <math>\overset{X}{\underset{x}{\forall}}~ \downharpoonleft x \in P \downharpoonright (x) ~=~ \downharpoonleft x \in Q \downharpoonright (x)</math>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10d~:~R7c}</math>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10d~:~R8c}</math>
 
|- style="height:20px"
 
|- style="height:20px"
 
| &nbsp;
 
| &nbsp;
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| &nbsp;
 
| &nbsp;
 
| <math>\operatorname{R10e.}</math>
 
| <math>\operatorname{R10e.}</math>
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| <math>\operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~\Leftrightarrow~ \upharpoonleft Q \upharpoonright (x))</math>
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| <math>\operatorname{Conj_x^X}~ (\downharpoonleft x \in P \downharpoonright (x) ~=~ \downharpoonleft x \in Q \downharpoonright (x))</math>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10e~:~R7d}</math>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10e~:~R8d}</math>
 
|- style="height:20px"
 
|- style="height:20px"
 
| &nbsp;
 
| &nbsp;
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| &nbsp;
 
| &nbsp;
 
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
 
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:60px"
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|- style="height:40px"
 
| &nbsp;
 
| &nbsp;
 
| <math>\operatorname{R10f.}</math>
 
| <math>\operatorname{R10f.}</math>
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| <math>\operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright (x) ~,~ \upharpoonleft Q \upharpoonright (x) ~\underline{))}</math>
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| <math>\operatorname{Conj_x^X}~ (\downharpoonleft x \in P \downharpoonright (x) ~\Leftrightarrow~ \downharpoonleft x \in Q \downharpoonright (x))</math>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10f~:~R7e}</math>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10f~:~R8e}</math>
 
|- style="height:20px"
 
|- style="height:20px"
 
| &nbsp;
 
| &nbsp;
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| &nbsp;
 
| &nbsp;
 
| <math>\operatorname{R10g.}</math>
 
| <math>\operatorname{R10g.}</math>
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| <math>\operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright ~\underline{))}^\$ (x)</math>
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| <math>\operatorname{Conj_x^X}~ \underline{((}~ \downharpoonleft x \in P \downharpoonright (x) ~,~ \downharpoonleft x \in Q \downharpoonright (x) ~\underline{))}</math>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10g~:~R7f}</math>
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| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10g~:~R8f}</math>
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|- style="height:20px"
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| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:40px"
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| &nbsp;
 +
| <math>\operatorname{R10h.}</math>
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| <math>\operatorname{Conj_x^X}~ \underline{((}~ \downharpoonleft x \in P \downharpoonright ~,~ \downharpoonleft x \in Q \downharpoonright ~\underline{))}^\$ (x)</math>
 +
| style="border-left:1px solid black; text-align:center" | <math>\operatorname{R10h~:~R8g}</math>
 
|}
 
|}
 
|}
 
|}
Jon Awbrey
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