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Theorem nelne1 2695
Description: Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)
Assertion
Ref Expression
nelne1  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  B  =/=  C )

Proof of Theorem nelne1
StepHypRef Expression
1 eleq2 2499 . . . 4  |-  ( B  =  C  ->  ( A  e.  B  <->  A  e.  C ) )
21biimpcd 217 . . 3  |-  ( A  e.  B  ->  ( B  =  C  ->  A  e.  C ) )
32necon3bd 2640 . 2  |-  ( A  e.  B  ->  ( -.  A  e.  C  ->  B  =/=  C ) )
43imp 420 1  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  B  =/=  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601
This theorem is referenced by:  difsnb  3942  fofinf1o  7390  fin23lem24  8207  fin23lem31  8228  ttukeylem7  8400  npomex  8878  lbspss  16159  islbs3  16232  lbsextlem4  16238  obslbs  16962  hauspwpwf1  18024  ppiltx  20965  ex-pss  21741  cntnevol  24587  rpnnen3lem  27116  lshpnelb  29856  osumcllem10N  30836  pexmidlem7N  30847  dochsnkrlem1  32341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-cleq 2431  df-clel 2434  df-ne 2603
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