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Theorem eldifsni 3736
Description: Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)
Assertion
Ref Expression
eldifsni  |-  ( A  e.  ( B  \  { C } )  ->  A  =/=  C )

Proof of Theorem eldifsni
StepHypRef Expression
1 eldifsn 3734 . 2  |-  ( A  e.  ( B  \  { C } )  <->  ( A  e.  B  /\  A  =/= 
C ) )
21simprbi 275 1  |-  ( A  e.  ( B  \  { C } )  ->  A  =/=  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2160    =/= wne 2360    \ cdif 3141   {csn 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-v 2754  df-dif 3146  df-sn 3613
This theorem is referenced by:  neldifsn  3737  suppssfv  6097  suppssov1  6098  elfi2  6989  fiuni  6995  fifo  6997  en2other2  7213  oddprm  12277  ringelnzr  13495  lgslem1  14798  lgseisenlem2  14848  m1lgs  14849
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