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Theorem en3lplem2 7671
Description: Lemma for en3lp 7672. (Contributed by Alan Sare, 28-Oct-2011.)
Assertion
Ref Expression
en3lplem2  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( x  i^i  { A ,  B ,  C }
)  =/=  (/) ) )
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem en3lplem2
StepHypRef Expression
1 en3lplem1 7670 . . . . 5  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  A  ->  ( x  i^i 
{ A ,  B ,  C } )  =/=  (/) ) )
2 en3lplem1 7670 . . . . . . . 8  |-  ( ( B  e.  C  /\  C  e.  A  /\  A  e.  B )  ->  ( x  =  B  ->  ( x  i^i 
{ B ,  C ,  A } )  =/=  (/) ) )
323comr 1161 . . . . . . 7  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  B  ->  ( x  i^i 
{ B ,  C ,  A } )  =/=  (/) ) )
43a1d 23 . . . . . 6  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( x  =  B  ->  (
x  i^i  { B ,  C ,  A }
)  =/=  (/) ) ) )
5 tprot 3899 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
65ineq2i 3539 . . . . . . . 8  |-  ( x  i^i  { A ,  B ,  C }
)  =  ( x  i^i  { B ,  C ,  A }
)
76neeq1i 2611 . . . . . . 7  |-  ( ( x  i^i  { A ,  B ,  C }
)  =/=  (/)  <->  ( x  i^i  { B ,  C ,  A } )  =/=  (/) )
87bicomi 194 . . . . . 6  |-  ( ( x  i^i  { B ,  C ,  A }
)  =/=  (/)  <->  ( x  i^i  { A ,  B ,  C } )  =/=  (/) )
94, 8syl8ib 223 . . . . 5  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( x  =  B  ->  (
x  i^i  { A ,  B ,  C }
)  =/=  (/) ) ) )
10 jao 499 . . . . 5  |-  ( ( x  =  A  -> 
( x  i^i  { A ,  B ,  C } )  =/=  (/) )  -> 
( ( x  =  B  ->  ( x  i^i  { A ,  B ,  C } )  =/=  (/) )  ->  ( ( x  =  A  \/  x  =  B )  ->  ( x  i^i  { A ,  B ,  C } )  =/=  (/) ) ) )
111, 9, 10sylsyld 54 . . . 4  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( ( x  =  A  \/  x  =  B )  ->  ( x  i^i  { A ,  B ,  C } )  =/=  (/) ) ) )
1211imp 419 . . 3  |-  ( ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A
)  /\  x  e.  { A ,  B ,  C } )  ->  (
( x  =  A  \/  x  =  B )  ->  ( x  i^i  { A ,  B ,  C } )  =/=  (/) ) )
13 en3lplem1 7670 . . . . . . 7  |-  ( ( C  e.  A  /\  A  e.  B  /\  B  e.  C )  ->  ( x  =  C  ->  ( x  i^i 
{ C ,  A ,  B } )  =/=  (/) ) )
14133coml 1160 . . . . . 6  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  C  ->  ( x  i^i 
{ C ,  A ,  B } )  =/=  (/) ) )
1514a1d 23 . . . . 5  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( x  =  C  ->  (
x  i^i  { C ,  A ,  B }
)  =/=  (/) ) ) )
16 tprot 3899 . . . . . . 7  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
1716ineq2i 3539 . . . . . 6  |-  ( x  i^i  { C ,  A ,  B }
)  =  ( x  i^i  { A ,  B ,  C }
)
1817neeq1i 2611 . . . . 5  |-  ( ( x  i^i  { C ,  A ,  B }
)  =/=  (/)  <->  ( x  i^i  { A ,  B ,  C } )  =/=  (/) )
1915, 18syl8ib 223 . . . 4  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( x  =  C  ->  (
x  i^i  { A ,  B ,  C }
)  =/=  (/) ) ) )
2019imp 419 . . 3  |-  ( ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A
)  /\  x  e.  { A ,  B ,  C } )  ->  (
x  =  C  -> 
( x  i^i  { A ,  B ,  C } )  =/=  (/) ) )
21 idd 22 . . . . . . 7  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  x  e. 
{ A ,  B ,  C } ) )
22 dftp2 3854 . . . . . . . 8  |-  { A ,  B ,  C }  =  { x  |  ( x  =  A  \/  x  =  B  \/  x  =  C ) }
2322eleq2i 2500 . . . . . . 7  |-  ( x  e.  { A ,  B ,  C }  <->  x  e.  { x  |  ( x  =  A  \/  x  =  B  \/  x  =  C ) } )
2421, 23syl6ib 218 . . . . . 6  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  x  e. 
{ x  |  ( x  =  A  \/  x  =  B  \/  x  =  C ) } ) )
25 abid 2424 . . . . . 6  |-  ( x  e.  { x  |  ( x  =  A  \/  x  =  B  \/  x  =  C ) }  <->  ( x  =  A  \/  x  =  B  \/  x  =  C ) )
2624, 25syl6ib 218 . . . . 5  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( x  =  A  \/  x  =  B  \/  x  =  C ) ) )
27 df-3or 937 . . . . 5  |-  ( ( x  =  A  \/  x  =  B  \/  x  =  C )  <->  ( ( x  =  A  \/  x  =  B )  \/  x  =  C ) )
2826, 27syl6ib 218 . . . 4  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( ( x  =  A  \/  x  =  B )  \/  x  =  C
) ) )
2928imp 419 . . 3  |-  ( ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A
)  /\  x  e.  { A ,  B ,  C } )  ->  (
( x  =  A  \/  x  =  B )  \/  x  =  C ) )
3012, 20, 29mpjaod 371 . 2  |-  ( ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A
)  /\  x  e.  { A ,  B ,  C } )  ->  (
x  i^i  { A ,  B ,  C }
)  =/=  (/) )
3130ex 424 1  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( x  i^i  { A ,  B ,  C }
)  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    \/ w3o 935    /\ w3a 936    = wceq 1652    e. wcel 1725   {cab 2422    =/= wne 2599    i^i cin 3319   (/)c0 3628   {ctp 3816
This theorem is referenced by:  en3lp  7672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-nul 3629  df-sn 3820  df-pr 3821  df-tp 3822
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