Theorem difsn 2455

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 2157	. . 3 |- (B \ {A}) (_ B
2	1	a1i 8	. 2 |- (-. A e. B -> (B \ {A}) (_ B)
3		nelneq 1553	. . . . . . 7 |- ((x e. B /\ -. A e. B) -> -. x = A)
4		df-ne 1579	. . . . . . 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 2453	. . . 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 2060	. 2 |- (-. A e. B -> B (_ (B \ {A}))
11	2, 10	eqssd 2069	1 |- (-. A e. B -> (B \ {A}) = B)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577   \ cdif 2034   (_ wss 2037  {csn 2399

This theorem is referenced by:  sspr 2466  clslp 7689

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-sn 2402  df-pr 2403

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