Theorem difsn 3574

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 3567	. . 3 |- ( x e. ( B \ { A } ) <-> ( x e. B /\ x =/= A ) )
2		simpl 107	. . . 4 |- ( ( x e. B /\ x =/= A ) -> x e. B )
3		eleq1 2150	. . . . . . . 8 |- ( x = A -> ( x e. B <-> A e. B ) )
4	3	biimpcd 157	. . . . . . 7 |- ( x e. B -> ( x = A -> A e. B ) )
5	4	necon3bd 2298	. . . . . 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 318	. . . 4 |- ( -. A e. B -> ( x e. B -> ( x e. B /\ x =/= A ) ) )
8	2, 7	impbid2 141	. . 3 |- ( -. A e. B -> ( ( x e. B /\ x =/= A ) <-> x e. B ) )
9	1, 8	syl5bb 190	. 2 |- ( -. A e. B -> ( x e. ( B \ { A } ) <-> x e. B ) )
10	9	eqrdv 2086	1 |- ( -. A e. B -> ( B \ { A } ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   = wceq 1289   e. wcel 1438   =/= wne 2255   \ cdif 2996   {csn 3446

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-dif 3001  df-sn 3452

This theorem is referenced by:  difsnb  3580  fisseneq  6640  dfn2  8684