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Theorem nelne1 2548
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 2357 . . . 4  |-  ( B  =  C  ->  ( A  e.  B  <->  A  e.  C ) )
21biimpcd 215 . . 3  |-  ( A  e.  B  ->  ( B  =  C  ->  A  e.  C ) )
32necon3bd 2496 . 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 1632    e. wcel 1696    =/= wne 2459
This theorem is referenced by:  difsnb  3773  fofinf1o  7153  fin23lem24  7964  fin23lem31  7985  ttukeylem7  8158  canth4  8285  npomex  8636  lbspss  15851  islbs3  15924  lbsextlem4  15930  obslbs  16646  hauspwpwf1  17698  ppiltx  20431  ex-pss  20831  cntnevol  23191  lppotos  26247  rpnnen3lem  27227  lshpnelb  29796  osumcllem10N  30776  pexmidlem7N  30787  dochsnkrlem1  32281
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-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-cleq 2289  df-clel 2292  df-ne 2461
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