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Theorem en3lplem1 7416
Description: Lemma for en3lp 7418. (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 957 . . 3  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  C  e.  A )
2 eleq2 2344 . . 3  |-  ( x  =  A  ->  ( C  e.  x  <->  C  e.  A ) )
31, 2syl5ibrcom 213 . 2  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  A  ->  C  e.  x
) )
4 tpid3g 3741 . . . . 5  |-  ( C  e.  A  ->  C  e.  { A ,  B ,  C } )
543ad2ant3 978 . . . 4  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  C  e.  { A ,  B ,  C }
)
6 inelcm 3509 . . . 4  |-  ( ( C  e.  x  /\  C  e.  { A ,  B ,  C }
)  ->  ( x  i^i  { A ,  B ,  C } )  =/=  (/) )
75, 6sylan2 460 . . 3  |-  ( ( C  e.  x  /\  ( A  e.  B  /\  B  e.  C  /\  C  e.  A
) )  ->  (
x  i^i  { A ,  B ,  C }
)  =/=  (/) )
87expcom 424 . 2  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( C  e.  x  ->  ( x  i^i  { A ,  B ,  C } )  =/=  (/) ) )
93, 8syld 40 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 934    = wceq 1623    e. wcel 1684    =/= wne 2446    i^i cin 3151   (/)c0 3455   {ctp 3642
This theorem is referenced by:  en3lplem2  7417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-nul 3456  df-sn 3646  df-pr 3647  df-tp 3648
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