Theorem dfss4 2827

Description: Subclass defined in terms of class difference. See comments under dfun2 2829. (The proof was shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem dfss4

Step	Hyp	Ref	Expression
1		sseqin2 2811	. 2 |- (A C_ B <-> (B i^i A) = A)
2		eldif 2609	. . . . . . 7 |- (x e. (B \ A) <-> (x e. B /\ -. x e. A))
3	2	notbii 204	. . . . . 6 |- (-. x e. (B \ A) <-> -. (x e. B /\ -. x e. A))
4	3	anbi2i 538	. . . . 5 |- ((x e. B /\ -. x e. (B \ A)) <-> (x e. B /\ -. (x e. B /\ -. x e. A)))
5		elin 2786	. . . . . 6 |- (x e. (B i^i A) <-> (x e. B /\ x e. A))
6		abai 537	. . . . . 6 |- ((x e. B /\ x e. A) <-> (x e. B /\ (x e. B -> x e. A)))
7		iman 256	. . . . . . 7 |- ((x e. B -> x e. A) <-> -. (x e. B /\ -. x e. A))
8	7	anbi2i 538	. . . . . 6 |- ((x e. B /\ (x e. B -> x e. A)) <-> (x e. B /\ -. (x e. B /\ -. x e. A)))
9	5, 6, 8	3bitri 194	. . . . 5 |- (x e. (B i^i A) <-> (x e. B /\ -. (x e. B /\ -. x e. A)))
10	4, 9	bitr4i 193	. . . 4 |- ((x e. B /\ -. x e. (B \ A)) <-> x e. (B i^i A))
11	10	difeqri 2727	. . 3 |- (B \ (B \ A)) = (B i^i A)
12	11	eqeq1i 1891	. 2 |- ((B \ (B \ A)) = A <-> (B i^i A) = A)
13	1, 12	bitr4i 193	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 is referenced by:  dfin4 2835  sbthlem3 5512  isopn2 8949  iincld 8955  ntrval2 8962  cmclsopn 8969  cmntrcld 8970  islp2 9023  subcld 10254  rcfpfillem6 14933  opncldf1 15402  opncldf3 15404  ntrcmp 15406  clscmp 15407  cldbnd 15410  clsun 15413  ufileu 15573  filufint 15574  ufilen 15579  filcon 15580  fcluscf 15612  flimfnfcls 15615  fdc 15812

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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