MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  en3lplem1 Unicode version

Theorem en3lplem1 7432
Description: Lemma for en3lp 7434. (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 2357 . . 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 3754 . . . . 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 3522 . . . 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 1632    e. wcel 1696    =/= wne 2459    i^i cin 3164   (/)c0 3468   {ctp 3655
This theorem is referenced by:  en3lplem2  7433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
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 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-nul 3469  df-sn 3659  df-pr 3660  df-tp 3661
  Copyright terms: Public domain W3C validator