Theorem difsn 3833

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 3822	. . 3 |- ( x e. ( B \ { A } ) <-> ( x e. B /\ x =/= A ) )
2		simpl 109	. . . 4 |- ( ( x e. B /\ x =/= A ) -> x e. B )
3		eleq1 2297	. . . . . . . 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 2457	. . . . . 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 2232	1 |- ( -. A e. B -> ( B \ { A } ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   =/= wne 2414   \ cdif 3210   {csn 3691

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-v 2817  df-dif 3215  df-sn 3697

This theorem is referenced by:  difsnb  3839  fisseneq  7197  dfn2  9514