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Theorem nelne1 2699
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 2502 . . . 4  |-  ( B  =  C  ->  ( A  e.  B  <->  A  e.  C ) )
21biimpcd 224 . . 3  |-  ( A  e.  B  ->  ( B  =  C  ->  A  e.  C ) )
32necon3bd 2643 . 2  |-  ( A  e.  B  ->  ( -.  A  e.  C  ->  B  =/=  C ) )
43imp 429 1  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  B  =/=  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-12 1797  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592  df-cleq 2434  df-clel 2437  df-ne 2606
This theorem is referenced by:  difsnb  4012  fofinf1o  7588  fin23lem24  8487  fin23lem31  8508  ttukeylem7  8680  npomex  9161  lbspss  17141  islbs3  17214  lbsextlem4  17220  obslbs  18114  hauspwpwf1  19519  ppiltx  22474  tglineneq  23001  ex-pss  23570  cntnevol  26578  fin2solem  28340  rpnnen3lem  29305  lvecpsslmod  30890  lshpnelb  32351  osumcllem10N  33331  pexmidlem7N  33342  dochsnkrlem1  34836
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