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Theorem nelne1 2535
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 2344 . . . 4  |-  ( B  =  C  ->  ( A  e.  B  <->  A  e.  C ) )
21biimpcd 215 . . 3  |-  ( A  e.  B  ->  ( B  =  C  ->  A  e.  C ) )
32necon3bd 2483 . 2  |-  ( A  e.  B  ->  ( -.  A  e.  C  ->  B  =/=  C ) )
43imp 418 1  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  B  =/=  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446
This theorem is referenced by:  difsneq  3757  fofinf1o  7137  fin23lem24  7948  fin23lem31  7969  ttukeylem7  8142  canth4  8269  npomex  8620  lbspss  15835  islbs3  15908  lbsextlem4  15914  obslbs  16630  hauspwpwf1  17682  ppiltx  20415  ex-pss  20815  cntnevol  23175  lppotos  26144  rpnnen3lem  27124  lshpnelb  29174  osumcllem10N  30154  pexmidlem7N  30165  dochsnkrlem1  31659
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-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-cleq 2276  df-clel 2279  df-ne 2448
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