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Theorem en3lplem1 7662
Description: Lemma for en3lp 7664. (Contributed by Alan Sare, 28-Oct-2011.)
Assertion
Ref Expression
en3lplem1  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  A  ->  ( x  i^i 
{ A ,  B ,  C } )  =/=  (/) ) )
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem en3lplem1
StepHypRef Expression
1 simp3 959 . . 3  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  C  e.  A )
2 eleq2 2496 . . 3  |-  ( x  =  A  ->  ( C  e.  x  <->  C  e.  A ) )
31, 2syl5ibrcom 214 . 2  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  A  ->  C  e.  x
) )
4 tpid3g 3911 . . . . 5  |-  ( C  e.  A  ->  C  e.  { A ,  B ,  C } )
543ad2ant3 980 . . . 4  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  C  e.  { A ,  B ,  C }
)
6 inelcm 3674 . . . 4  |-  ( ( C  e.  x  /\  C  e.  { A ,  B ,  C }
)  ->  ( x  i^i  { A ,  B ,  C } )  =/=  (/) )
75, 6sylan2 461 . . 3  |-  ( ( C  e.  x  /\  ( A  e.  B  /\  B  e.  C  /\  C  e.  A
) )  ->  (
x  i^i  { A ,  B ,  C }
)  =/=  (/) )
87expcom 425 . 2  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( C  e.  x  ->  ( x  i^i  { A ,  B ,  C } )  =/=  (/) ) )
93, 8syld 42 1  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  A  ->  ( x  i^i 
{ A ,  B ,  C } )  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598    i^i cin 3311   (/)c0 3620   {ctp 3808
This theorem is referenced by:  en3lplem2  7663
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-nul 3621  df-sn 3812  df-pr 3813  df-tp 3814
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