Theorem eldifsni 3802

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 3800	. 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 2202   =/= wne 2402   \ cdif 3197   {csn 3669

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-v 2804  df-dif 3202  df-sn 3675

This theorem is referenced by:  neldifsn  3803  suppssfv  6230  suppssov1  6231  elfi2  7170  fiuni  7176  fifo  7178  en2other2  7406  oddprm  12831  ringelnzr  14200  lgslem1  15728  lgseisenlem2  15799  lgseisenlem4  15801  lgseisen  15802  lgsquadlem1  15805  lgsquad2  15811  m1lgs  15813