Theorem dfss4OLD 2828

Description: Subclass defined in terms of class difference. See comments under dfun2 2829.

Assertion

Ref	Expression
dfss4OLD	|- (A C_ B <-> (B \ (B \ A)) = A)

Proof of Theorem dfss4OLD

Step	Hyp	Ref	Expression
1		sseqin2 2811	. 2 |- (A C_ B <-> (B i^i A) = A)
2		abai 537	. . . . . 6 |- ((x e. B /\ x e. A) <-> (x e. B /\ (x e. B -> x e. A)))
3		iman 256	. . . . . . 7 |- ((x e. B -> x e. A) <-> -. (x e. B /\ -. x e. A))
4	3	anbi2i 538	. . . . . 6 |- ((x e. B /\ (x e. B -> x e. A)) <-> (x e. B /\ -. (x e. B /\ -. x e. A)))
5	2, 4	bitri 190	. . . . 5 |- ((x e. B /\ x e. A) <-> (x e. B /\ -. (x e. B /\ -. x e. A)))
6		elin 2786	. . . . 5 |- (x e. (B i^i A) <-> (x e. B /\ x e. A))
7		eldif 2609	. . . . . 6 |- (x e. (B \ (B \ A)) <-> (x e. B /\ -. x e. (B \ A)))
8		eldif 2609	. . . . . . . 8 |- (x e. (B \ A) <-> (x e. B /\ -. x e. A))
9	8	notbii 204	. . . . . . 7 |- (-. x e. (B \ A) <-> -. (x e. B /\ -. x e. A))
10	9	anbi2i 538	. . . . . 6 |- ((x e. B /\ -. x e. (B \ A)) <-> (x e. B /\ -. (x e. B /\ -. x e. A)))
11	7, 10	bitri 190	. . . . 5 |- (x e. (B \ (B \ A)) <-> (x e. B /\ -. (x e. B /\ -. x e. A)))
12	5, 6, 11	3bitr4i 200	. . . 4 |- (x e. (B i^i A) <-> x e. (B \ (B \ A)))
13	12	eqriv 1881	. . 3 |- (B i^i A) = (B \ (B \ A))
14	13	eqeq1i 1891	. 2 |- ((B i^i A) = A <-> (B \ (B \ A)) = A)
15	1, 14	bitri 190	1 |- (A C_ B <-> (B \ (B \ A)) = A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   \ cdif 2590   i^i cin 2592   C_ wss 2593

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605

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