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Theorem en3lplem2 8035
Description: Lemma for en3lp 8036. (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 8034 . . . . 5  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  A  ->  ( x  i^i 
{ A ,  B ,  C } )  =/=  (/) ) )
2 en3lplem1 8034 . . . . . . . 8  |-  ( ( B  e.  C  /\  C  e.  A  /\  A  e.  B )  ->  ( x  =  B  ->  ( x  i^i 
{ B ,  C ,  A } )  =/=  (/) ) )
323comr 1205 . . . . . . 7  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  B  ->  ( x  i^i 
{ B ,  C ,  A } )  =/=  (/) ) )
43a1d 25 . . . . . 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 4110 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
65ineq2i 3682 . . . . . . . 8  |-  ( x  i^i  { A ,  B ,  C }
)  =  ( x  i^i  { B ,  C ,  A }
)
76neeq1i 2728 . . . . . . 7  |-  ( ( x  i^i  { A ,  B ,  C }
)  =/=  (/)  <->  ( x  i^i  { B ,  C ,  A } )  =/=  (/) )
87bicomi 202 . . . . . 6  |-  ( ( x  i^i  { B ,  C ,  A }
)  =/=  (/)  <->  ( x  i^i  { A ,  B ,  C } )  =/=  (/) )
94, 8syl8ib 231 . . . . 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 512 . . . . 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 56 . . . 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 429 . . 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 8034 . . . . . . 7  |-  ( ( C  e.  A  /\  A  e.  B  /\  B  e.  C )  ->  ( x  =  C  ->  ( x  i^i 
{ C ,  A ,  B } )  =/=  (/) ) )
14133coml 1204 . . . . . 6  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  C  ->  ( x  i^i 
{ C ,  A ,  B } )  =/=  (/) ) )
1514a1d 25 . . . . 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 4110 . . . . . . 7  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
1716ineq2i 3682 . . . . . 6  |-  ( x  i^i  { C ,  A ,  B }
)  =  ( x  i^i  { A ,  B ,  C }
)
1817neeq1i 2728 . . . . 5  |-  ( ( x  i^i  { C ,  A ,  B }
)  =/=  (/)  <->  ( x  i^i  { A ,  B ,  C } )  =/=  (/) )
1915, 18syl8ib 231 . . . 4  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( x  =  C  ->  (
x  i^i  { A ,  B ,  C }
)  =/=  (/) ) ) )
2019imp 429 . . 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 24 . . . . . . 7  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  x  e. 
{ A ,  B ,  C } ) )
22 dftp2 4060 . . . . . . . 8  |-  { A ,  B ,  C }  =  { x  |  ( x  =  A  \/  x  =  B  \/  x  =  C ) }
2322eleq2i 2521 . . . . . . 7  |-  ( x  e.  { A ,  B ,  C }  <->  x  e.  { x  |  ( x  =  A  \/  x  =  B  \/  x  =  C ) } )
2421, 23syl6ib 226 . . . . . 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 2430 . . . . . 6  |-  ( x  e.  { x  |  ( x  =  A  \/  x  =  B  \/  x  =  C ) }  <->  ( x  =  A  \/  x  =  B  \/  x  =  C ) )
2624, 25syl6ib 226 . . . . 5  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( x  =  A  \/  x  =  B  \/  x  =  C ) ) )
27 df-3or 975 . . . . 5  |-  ( ( x  =  A  \/  x  =  B  \/  x  =  C )  <->  ( ( x  =  A  \/  x  =  B )  \/  x  =  C ) )
2826, 27syl6ib 226 . . . 4  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( ( x  =  A  \/  x  =  B )  \/  x  =  C
) ) )
2928imp 429 . . 3  |-  ( ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A
)  /\  x  e.  { A ,  B ,  C } )  ->  (
( x  =  A  \/  x  =  B )  \/  x  =  C ) )
3012, 20, 29mpjaod 381 . 2  |-  ( ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A
)  /\  x  e.  { A ,  B ,  C } )  ->  (
x  i^i  { A ,  B ,  C }
)  =/=  (/) )
3130ex 434 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 setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 973    /\ w3a 974    = wceq 1383    e. wcel 1804   {cab 2428    =/= wne 2638    i^i cin 3460   (/)c0 3770   {ctp 4018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-nul 3771  df-sn 4015  df-pr 4017  df-tp 4019
This theorem is referenced by:  en3lp  8036
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