Theorem eldifsn 2462

Description: Membership in a set with an element removed.

Assertion

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

Proof of Theorem eldifsn

Step	Hyp	Ref	Expression
1		eldif 2057	. 2 |- (A e. (B \ {C}) <-> (A e. B /\ -. A e. {C}))
2		elsncg 2430	. . . 4 |- (A e. B -> (A e. {C} <-> A = C))
3	2	necon3bbid 1600	. . 3 |- (A e. B -> (-. A e. {C} <-> A =/= C))
4	3	pm5.32i 645	. 2 |- ((A e. B /\ -. A e. {C}) <-> (A e. B /\ A =/= C))
5	1, 4	bitr 173	1 |- (A e. (B \ {C}) <-> (A e. B /\ A =/= C))

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   e. wcel 958   =/= wne 1585   \ cdif 2044  {csn 2409

This theorem is referenced by:  difsn 2464  difprsn 2465  onmindif2 3061  limenpsi 4505  kmlem3 4767  kmlem4 4768  kmlem11 4775  elni 5004  mulnzcnopr 5702  divval 5704  dfn2 6112  seq1lem2 6310  eff2 7370  acdc5lem1 7491  ruclem8 7517  bcthlem30 8028  ablmul 8131  mulid 8132  effoi 8745  logclt 8758  eflogt 8760  logeftb 8764  blkssatm 10767

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-sn 2412  df-pr 2413

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