Theorem difsn 3759

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 3749	. . 3 |- ( x e. ( B \ { A } ) <-> ( x e. B /\ x =/= A ) )
2		simpl 109	. . . 4 |- ( ( x e. B /\ x =/= A ) -> x e. B )
3		eleq1 2259	. . . . . . . 8 |- ( x = A -> ( x e. B <-> A e. B ) )
4	3	biimpcd 159	. . . . . . 7 |- ( x e. B -> ( x = A -> A e. B ) )
5	4	necon3bd 2410	. . . . . 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 325	. . . 4 |- ( -. A e. B -> ( x e. B -> ( x e. B /\ x =/= A ) ) )
8	2, 7	impbid2 143	. . 3 |- ( -. A e. B -> ( ( x e. B /\ x =/= A ) <-> x e. B ) )
9	1, 8	bitrid 192	. 2 |- ( -. A e. B -> ( x e. ( B \ { A } ) <-> x e. B ) )
10	9	eqrdv 2194	1 |- ( -. A e. B -> ( B \ { A } ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   =/= wne 2367   \ cdif 3154   {csn 3622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-v 2765  df-dif 3159  df-sn 3628

This theorem is referenced by:  difsnb  3765  fisseneq  6995  dfn2  9262