Theorem difsn 4328

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 4317	. . 3 |- ( x e. ( B \ { A } ) <-> ( x e. B /\ x =/= A ) )
2		simpl 473	. . . 4 |- ( ( x e. B /\ x =/= A ) -> x e. B )
3		nelelne 2892	. . . . 5 |- ( -. A e. B -> ( x e. B -> x =/= A ) )
4	3	ancld 576	. . . 4 |- ( -. A e. B -> ( x e. B -> ( x e. B /\ x =/= A ) ) )
5	2, 4	impbid2 216	. . 3 |- ( -. A e. B -> ( ( x e. B /\ x =/= A ) <-> x e. B ) )
6	1, 5	syl5bb 272	. 2 |- ( -. A e. B -> ( x e. ( B \ { A } ) <-> x e. B ) )
7	6	eqrdv 2620	1 |- ( -. A e. B -> ( B \ { A } ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-sn 4178

This theorem is referenced by:  difsnb  4337  difsnexi  6970  domdifsn  8043  domunsncan  8060  frfi  8205  infdifsn  8554  dfn2  11305  clslp  20952  xrge00  29686  lindsenlbs  33404  poimirlem2  33411  poimirlem4  33413  poimirlem6  33415  poimirlem7  33416  poimirlem8  33417  poimirlem19  33428  poimirlem23  33432  supxrmnf2  39660  infxrpnf2  39693  dvmptfprodlem  40159  hoiprodp1  40802