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Theorem en3lplem1 8067
Description: Lemma for en3lp 8069. (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 1007 . . 3  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  C  e.  A )
2 eleq2 2490 . . 3  |-  ( x  =  A  ->  ( C  e.  x  <->  C  e.  A ) )
31, 2syl5ibrcom 225 . 2  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  A  ->  C  e.  x
) )
4 tpid3g 4053 . . . . 5  |-  ( C  e.  A  ->  C  e.  { A ,  B ,  C } )
543ad2ant3 1028 . . . 4  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  C  e.  { A ,  B ,  C }
)
6 inelcm 3787 . . . 4  |-  ( ( C  e.  x  /\  C  e.  { A ,  B ,  C }
)  ->  ( x  i^i  { A ,  B ,  C } )  =/=  (/) )
75, 6sylan2 476 . . 3  |-  ( ( C  e.  x  /\  ( A  e.  B  /\  B  e.  C  /\  C  e.  A
) )  ->  (
x  i^i  { A ,  B ,  C }
)  =/=  (/) )
87expcom 436 . 2  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( C  e.  x  ->  ( x  i^i  { A ,  B ,  C } )  =/=  (/) ) )
93, 8syld 45 1  |-  ( ( A  e.  B  /\  B  e.  C  /\  C  e.  A )  ->  ( x  =  A  ->  ( x  i^i 
{ A ,  B ,  C } )  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2594    i^i cin 3373   (/)c0 3699   {ctp 3940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403
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 1658  df-nf 1662  df-sb 1791  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-v 3019  df-dif 3377  df-un 3379  df-in 3381  df-nul 3700  df-sn 3937  df-pr 3939  df-tp 3941
This theorem is referenced by:  en3lplem2  8068
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