Theorem eldifsni 3797

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

Step	Hyp	Ref	Expression
1		eldifsn 3795	. 2 |- ( A e. ( B \ { C } ) <-> ( A e. B /\ A =/= C ) )
2	1	simprbi 275	1 |- ( A e. ( B \ { C } ) -> A =/= C )

Syntax hints:   -> wi 4   e. wcel 2200   =/= wne 2400   \ cdif 3194   {csn 3666

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-v 2801  df-dif 3199  df-sn 3672

This theorem is referenced by:  neldifsn  3798  suppssfv  6220  suppssov1  6221  elfi2  7150  fiuni  7156  fifo  7158  en2other2  7385  oddprm  12798  ringelnzr  14167  lgslem1  15695  lgseisenlem2  15766  lgseisenlem4  15768  lgseisen  15769  lgsquadlem1  15772  lgsquad2  15778  m1lgs  15780