Theorem eldifsni 3577

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 3575	. 2 |- ( A e. ( B \ { C } ) <-> ( A e. B /\ A =/= C ) )
2	1	simprbi 270	1 |- ( A e. ( B \ { C } ) -> A =/= C )

Syntax hints:   -> wi 4   e. wcel 1439   =/= wne 2256   \ cdif 2999   {csn 3452

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-v 2624  df-dif 3004  df-sn 3458

This theorem is referenced by:  neldifsn  3578  suppssfv  5868  suppssov1  5869  en2other2  6885