Theorem difsn 2468

Description: An element not in a set can be removed without affecting the set.

Assertion

Ref	Expression
difsn	|- (-. A e. B -> (B \ {A}) = B)

Proof of Theorem difsn

Step	Hyp	Ref	Expression
1		difss 2170	. . 3 |- (B \ {A}) (_ B
2	1	a1i 8	. 2 |- (-. A e. B -> (B \ {A}) (_ B)
3		nelneq 1564	. . . . . . 7 |- ((x e. B /\ -. A e. B) -> -. x = A)
4		df-ne 1590	. . . . . . 7 |- (x =/= A <-> -. x = A)
5	3, 4	sylibr 200	. . . . . 6 |- ((x e. B /\ -. A e. B) -> x =/= A)
6	5	expcom 374	. . . . 5 |- (-. A e. B -> (x e. B -> x =/= A))
7	6	ancld 298	. . . 4 |- (-. A e. B -> (x e. B -> (x e. B /\ x =/= A)))
8		eldifsn 2466	. . . 4 |- (x e. (B \ {A}) <-> (x e. B /\ x =/= A))
9	7, 8	syl6ibr 213	. . 3 |- (-. A e. B -> (x e. B -> x e. (B \ {A})))
10	9	ssrdv 2073	. 2 |- (-. A e. B -> B (_ (B \ {A}))
11	2, 10	eqssd 2082	1 |- (-. A e. B -> (B \ {A}) = B)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960   =/= wne 1588   \ cdif 2047   (_ wss 2050  {csn 2413

This theorem is referenced by:  sspr 2479  clslp 7745

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-sn 2416  df-pr 2417

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