Theorem eldifsn 3495

Description: Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)

Assertion

Ref	Expression
eldifsn	|- ( A e. ( B \ { C } ) <-> ( A e. B /\ A =/= C ) )

Proof of Theorem eldifsn

Step	Hyp	Ref	Expression
1		eldif 2927	. 2 |- ( A e. ( B \ { C } ) <-> ( A e. B /\ -. A e. { C } ) )
2		elsng 3390	. . . 4 |- ( A e. B -> ( A e. { C } <-> A = C ) )
3	2	necon3bbid 2245	. . 3 |- ( A e. B -> ( -. A e. { C } <-> A =/= C ) )
4	3	pm5.32i 427	. 2 |- ( ( A e. B /\ -. A e. { C } ) <-> ( A e. B /\ A =/= C ) )
5	1, 4	bitri 173	1 |- ( A e. ( B \ { C } ) <-> ( A e. B /\ A =/= C ) )

Syntax hints:   -. wn 3   /\ wa 97   <-> wb 98   e. wcel 1393   =/= wne 2204   \ cdif 2914   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-sn 3381

This theorem is referenced by:  eldifsni  3496  rexdifsn  3499  difsn  3501  fnniniseg2  5290  rexsupp  5291  suppssfv  5708  suppssov1  5709  dif1o  6021  fidifsnen  6331  elni  6406  divvalap  7653  elnnne0  8195  divfnzn  8556