Theorem difsn 3845

Description: An element not in a set can be removed without affecting the set. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
difsn	|- (-. A e. B -> (B \ {A}) = B)

Proof of Theorem difsn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifsn 3839	. . 3 |- (x e. (B \ {A}) <-> (x e. B /\ x =/= A))
2		simpl 443	. . . 4 |- ((x e. B /\ x =/= A) -> x e. B)
3		eleq1 2413	. . . . . . . 8 |- (x = A -> (x e. B <-> A e. B))
4	3	biimpcd 215	. . . . . . 7 |- (x e. B -> (x = A -> A e. B))
5	4	necon3bd 2553	. . . . . 6 |- (x e. B -> (-. A e. B -> x =/= A))
6	5	com12 27	. . . . 5 |- (-. A e. B -> (x e. B -> x =/= A))
7	6	ancld 536	. . . 4 |- (-. A e. B -> (x e. B -> (x e. B /\ x =/= A)))
8	2, 7	impbid2 195	. . 3 |- (-. A e. B -> ((x e. B /\ x =/= A) <-> x e. B))
9	1, 8	syl5bb 248	. 2 |- (-. A e. B -> (x e. (B \ {A}) <-> x e. B))
10	9	eqrdv 2351	1 |- (-. A e. B -> (B \ {A}) = B)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2516   \ cdif 3206  {csn 3737

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-sn 3741

This theorem is referenced by:  difsnb  3850  adj11  3889  nnsucelrlem3  4426  nnsucelr  4428  ssfin  4470