Theorem difsn 3689

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 3682	. . 3 |- ( x e. ( B \ { A } ) <-> ( x e. B /\ x =/= A ) )
2		simpl 108	. . . 4 |- ( ( x e. B /\ x =/= A ) -> x e. B )
3		eleq1 2217	. . . . . . . 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 2367	. . . . . 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 2152	1 |- ( -. A e. B -> ( B \ { A } ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 2125   =/= wne 2324   \ cdif 3095   {csn 3556

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-v 2711  df-dif 3100  df-sn 3562

This theorem is referenced by:  difsnb  3695  fisseneq  6865  dfn2  9082