Theorem dfss4 3489

Description: Subclass defined in terms of class difference. See comments under dfun2 3490. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem dfss4

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseqin2 3474	. 2 |- (A C_ B <-> (B i^i A) = A)
2		eldif 3221	. . . . . . 7 |- (x e. (B \ A) <-> (x e. B /\ -. x e. A))
3	2	notbii 287	. . . . . 6 |- (-. x e. (B \ A) <-> -. (x e. B /\ -. x e. A))
4	3	anbi2i 675	. . . . 5 |- ((x e. B /\ -. x e. (B \ A)) <-> (x e. B /\ -. (x e. B /\ -. x e. A)))
5		elin 3219	. . . . . 6 |- (x e. (B i^i A) <-> (x e. B /\ x e. A))
6		abai 770	. . . . . 6 |- ((x e. B /\ x e. A) <-> (x e. B /\ (x e. B -> x e. A)))
7		iman 413	. . . . . . 7 |- ((x e. B -> x e. A) <-> -. (x e. B /\ -. x e. A))
8	7	anbi2i 675	. . . . . 6 |- ((x e. B /\ (x e. B -> x e. A)) <-> (x e. B /\ -. (x e. B /\ -. x e. A)))
9	5, 6, 8	3bitri 262	. . . . 5 |- (x e. (B i^i A) <-> (x e. B /\ -. (x e. B /\ -. x e. A)))
10	4, 9	bitr4i 243	. . . 4 |- ((x e. B /\ -. x e. (B \ A)) <-> x e. (B i^i A))
11	10	difeqri 3387	. . 3 |- (B \ (B \ A)) = (B i^i A)
12	11	eqeq1i 2360	. 2 |- ((B \ (B \ A)) = A <-> (B i^i A) = A)
13	1, 12	bitr4i 243	1 |- (A C_ B <-> (B \ (B \ A)) = A)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710   \ cdif 3206   i^i cin 3208   C_ wss 3257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259

This theorem is referenced by:  dfin4  3495