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Theorem en3lplem2 8120
Description: Lemma for en3lp 8121. (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 8119 . . . . 5  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  A  ->  ( x  i^i 
{ A ,  B ,  C } )  =/=  (/) ) )
2 en3lplem1 8119 . . . . . . . 8  |-  ( ( B  e.  C  /\  C  e.  A  /\  A  e.  B )  ->  ( x  =  B  ->  ( x  i^i 
{ B ,  C ,  A } )  =/=  (/) ) )
323comr 1213 . . . . . . 7  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  B  ->  ( x  i^i 
{ B ,  C ,  A } )  =/=  (/) ) )
43a1d 26 . . . . . 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 4098 . . . . . . . . 9  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
65ineq2i 3667 . . . . . . . 8  |-  ( x  i^i  { A ,  B ,  C }
)  =  ( x  i^i  { B ,  C ,  A }
)
76neeq1i 2716 . . . . . . 7  |-  ( ( x  i^i  { A ,  B ,  C }
)  =/=  (/)  <->  ( x  i^i  { B ,  C ,  A } )  =/=  (/) )
87bicomi 205 . . . . . 6  |-  ( ( x  i^i  { B ,  C ,  A }
)  =/=  (/)  <->  ( x  i^i  { A ,  B ,  C } )  =/=  (/) )
94, 8syl8ib 234 . . . . 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 514 . . . . 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 58 . . . 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 430 . . 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 8119 . . . . . . 7  |-  ( ( C  e.  A  /\  A  e.  B  /\  B  e.  C )  ->  ( x  =  C  ->  ( x  i^i 
{ C ,  A ,  B } )  =/=  (/) ) )
14133coml 1212 . . . . . 6  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  C  ->  ( x  i^i 
{ C ,  A ,  B } )  =/=  (/) ) )
1514a1d 26 . . . . 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 4098 . . . . . . 7  |-  { C ,  A ,  B }  =  { A ,  B ,  C }
1716ineq2i 3667 . . . . . 6  |-  ( x  i^i  { C ,  A ,  B }
)  =  ( x  i^i  { A ,  B ,  C }
)
1817neeq1i 2716 . . . . 5  |-  ( ( x  i^i  { C ,  A ,  B }
)  =/=  (/)  <->  ( x  i^i  { A ,  B ,  C } )  =/=  (/) )
1915, 18syl8ib 234 . . . 4  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( x  =  C  ->  (
x  i^i  { A ,  B ,  C }
)  =/=  (/) ) ) )
2019imp 430 . . 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 25 . . . . . . 7  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  x  e. 
{ A ,  B ,  C } ) )
22 dftp2 4049 . . . . . . . 8  |-  { A ,  B ,  C }  =  { x  |  ( x  =  A  \/  x  =  B  \/  x  =  C ) }
2322eleq2i 2507 . . . . . . 7  |-  ( x  e.  { A ,  B ,  C }  <->  x  e.  { x  |  ( x  =  A  \/  x  =  B  \/  x  =  C ) } )
2421, 23syl6ib 229 . . . . . 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 2416 . . . . . 6  |-  ( x  e.  { x  |  ( x  =  A  \/  x  =  B  \/  x  =  C ) }  <->  ( x  =  A  \/  x  =  B  \/  x  =  C ) )
2624, 25syl6ib 229 . . . . 5  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( x  =  A  \/  x  =  B  \/  x  =  C ) ) )
27 df-3or 983 . . . . 5  |-  ( ( x  =  A  \/  x  =  B  \/  x  =  C )  <->  ( ( x  =  A  \/  x  =  B )  \/  x  =  C ) )
2826, 27syl6ib 229 . . . 4  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  e.  { A ,  B ,  C }  ->  ( ( x  =  A  \/  x  =  B )  \/  x  =  C
) ) )
2928imp 430 . . 3  |-  ( ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A
)  /\  x  e.  { A ,  B ,  C } )  ->  (
( x  =  A  \/  x  =  B )  \/  x  =  C ) )
3012, 20, 29mpjaod 382 . 2  |-  ( ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A
)  /\  x  e.  { A ,  B ,  C } )  ->  (
x  i^i  { A ,  B ,  C }
)  =/=  (/) )
3130ex 435 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 369    /\ wa 370    \/ w3o 981    /\ w3a 982    = wceq 1437    e. wcel 1870   {cab 2414    =/= wne 2625    i^i cin 3441   (/)c0 3767   {ctp 4006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-nul 3768  df-sn 4003  df-pr 4005  df-tp 4007
This theorem is referenced by:  en3lp  8121
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