Theorem difsn 3717

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 3710	. . 3 |- ( x e. ( B \ { A } ) <-> ( x e. B /\ x =/= A ) )
2		simpl 108	. . . 4 |- ( ( x e. B /\ x =/= A ) -> x e. B )
3		eleq1 2233	. . . . . . . 8 |- ( x = A -> ( x e. B <-> A e. B ) )
4	3	biimpcd 158	. . . . . . 7 |- ( x e. B -> ( x = A -> A e. B ) )
5	4	necon3bd 2383	. . . . . 6 |- ( x e. B -> ( -. A e. B -> x =/= A ) )
6	5	com12 30	. . . . 5 |- ( -. A e. B -> ( x e. B -> x =/= A ) )
7	6	ancld 323	. . . 4 |- ( -. A e. B -> ( x e. B -> ( x e. B /\ x =/= A ) ) )
8	2, 7	impbid2 142	. . 3 |- ( -. A e. B -> ( ( x e. B /\ x =/= A ) <-> x e. B ) )
9	1, 8	syl5bb 191	. 2 |- ( -. A e. B -> ( x e. ( B \ { A } ) <-> x e. B ) )
10	9	eqrdv 2168	1 |- ( -. A e. B -> ( B \ { A } ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   =/= wne 2340   \ cdif 3118   {csn 3583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-v 2732  df-dif 3123  df-sn 3589

This theorem is referenced by:  difsnb  3723  fisseneq  6909  dfn2  9148