Theorem dfss4 2246

Description: Subclass defined in terms of class difference. See comments under dfun2 2247.

Assertion

Ref	Expression
dfss4	|- (A (_ B <-> (B \ (B \ A)) = A)

Proof of Theorem dfss4

Step	Hyp	Ref	Expression
1		sseqin2 2233	. 2 |- (A (_ B <-> (B i^i A) = A)
2		abai 481	. . . . . 6 |- ((x e. B /\ x e. A) <-> (x e. B /\ (x e. B -> x e. A)))
3		iman 237	. . . . . . 7 |- ((x e. B -> x e. A) <-> -. (x e. B /\ -. x e. A))
4	3	anbi2i 482	. . . . . 6 |- ((x e. B /\ (x e. B -> x e. A)) <-> (x e. B /\ -. (x e. B /\ -. x e. A)))
5	2, 4	bitr 173	. . . . 5 |- ((x e. B /\ x e. A) <-> (x e. B /\ -. (x e. B /\ -. x e. A)))
6		elin 2211	. . . . 5 |- (x e. (B i^i A) <-> (x e. B /\ x e. A))
7		eldif 2061	. . . . . 6 |- (x e. (B \ (B \ A)) <-> (x e. B /\ -. x e. (B \ A)))
8		eldif 2061	. . . . . . . 8 |- (x e. (B \ A) <-> (x e. B /\ -. x e. A))
9	8	negbii 187	. . . . . . 7 |- (-. x e. (B \ A) <-> -. (x e. B /\ -. x e. A))
10	9	anbi2i 482	. . . . . 6 |- ((x e. B /\ -. x e. (B \ A)) <-> (x e. B /\ -. (x e. B /\ -. x e. A)))
11	7, 10	bitr 173	. . . . 5 |- (x e. (B \ (B \ A)) <-> (x e. B /\ -. (x e. B /\ -. x e. A)))
12	5, 6, 11	3bitr4 183	. . . 4 |- (x e. (B i^i A) <-> x e. (B \ (B \ A)))
13	12	eqriv 1477	. . 3 |- (B i^i A) = (B \ (B \ A))
14	13	eqeq1i 1485	. 2 |- ((B i^i A) = A <-> (B \ (B \ A)) = A)
15	1, 14	bitr 173	1 |- (A (_ B <-> (B \ (B \ A)) = A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960   \ cdif 2048   i^i cin 2050   (_ wss 2051

This theorem is referenced by:  dfin4 2252  sbthlem3 4456  isopn2 7677  iincld 7683  ntrval2 7690  cmclsopn 7697  cmntrcld 7698  islp2 7751  rcfpfillem6 10576

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-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-dif 2053  df-in 2055  df-ss 2057

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