Theorem dfss4 2213

Description: Subclass defined in terms of class difference. See comments under dfun2 2214.

Assertion

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

Proof of Theorem dfss4

Step	Hyp	Ref	Expression
1		sseqin2 2200	. 2 |- (A (_ B <-> (B i^i A) = A)
2		abai 478	. . . . . 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 479	. . . . . 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 2178	. . . . 5 |- (x e. (B i^i A) <-> (x e. B /\ x e. A))
7		eldif 2028	. . . . . 6 |- (x e. (B \ (B \ A)) <-> (x e. B /\ -. x e. (B \ A)))
8		eldif 2028	. . . . . . . 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 479	. . . . . 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 1451	. . 3 |- (B i^i A) = (B \ (B \ A))
14	13	eqeq1i 1458	. 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 1099   e. wcel 1105   \ cdif 2015   i^i cin 2017   (_ wss 2018

This theorem is referenced by:  dfin4 2219  sbthlem3 4383  isopn2 7566  iincld 7572  ntrval2 7579  cmclsopn 7586  cmntrcld 7587  islp2 7636

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-v 1787  df-dif 2020  df-in 2022  df-ss 2024

Copyright terms: Public domain