Theorem ssconb 3400

Description: Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)

Assertion

Ref	Expression
ssconb	|- ((A C_ C /\ B C_ C) -> (A C_ (C \ B) <-> B C_ (C \ A)))

Proof of Theorem ssconb

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3268	. . . . . . 7 |- (A C_ C -> (x e. A -> x e. C))
2		ssel 3268	. . . . . . 7 |- (B C_ C -> (x e. B -> x e. C))
3		pm5.1 830	. . . . . . 7 |- (((x e. A -> x e. C) /\ (x e. B -> x e. C)) -> ((x e. A -> x e. C) <-> (x e. B -> x e. C)))
4	1, 2, 3	syl2an 463	. . . . . 6 |- ((A C_ C /\ B C_ C) -> ((x e. A -> x e. C) <-> (x e. B -> x e. C)))
5		con2b 324	. . . . . . 7 |- ((x e. A -> -. x e. B) <-> (x e. B -> -. x e. A))
6	5	a1i 10	. . . . . 6 |- ((A C_ C /\ B C_ C) -> ((x e. A -> -. x e. B) <-> (x e. B -> -. x e. A)))
7	4, 6	anbi12d 691	. . . . 5 |- ((A C_ C /\ B C_ C) -> (((x e. A -> x e. C) /\ (x e. A -> -. x e. B)) <-> ((x e. B -> x e. C) /\ (x e. B -> -. x e. A))))
8		jcab 833	. . . . 5 |- ((x e. A -> (x e. C /\ -. x e. B)) <-> ((x e. A -> x e. C) /\ (x e. A -> -. x e. B)))
9		jcab 833	. . . . 5 |- ((x e. B -> (x e. C /\ -. x e. A)) <-> ((x e. B -> x e. C) /\ (x e. B -> -. x e. A)))
10	7, 8, 9	3bitr4g 279	. . . 4 |- ((A C_ C /\ B C_ C) -> ((x e. A -> (x e. C /\ -. x e. B)) <-> (x e. B -> (x e. C /\ -. x e. A))))
11		eldif 3222	. . . . 5 |- (x e. (C \ B) <-> (x e. C /\ -. x e. B))
12	11	imbi2i 303	. . . 4 |- ((x e. A -> x e. (C \ B)) <-> (x e. A -> (x e. C /\ -. x e. B)))
13		eldif 3222	. . . . 5 |- (x e. (C \ A) <-> (x e. C /\ -. x e. A))
14	13	imbi2i 303	. . . 4 |- ((x e. B -> x e. (C \ A)) <-> (x e. B -> (x e. C /\ -. x e. A)))
15	10, 12, 14	3bitr4g 279	. . 3 |- ((A C_ C /\ B C_ C) -> ((x e. A -> x e. (C \ B)) <-> (x e. B -> x e. (C \ A))))
16	15	albidv 1625	. 2 |- ((A C_ C /\ B C_ C) -> (A.x(x e. A -> x e. (C \ B)) <-> A.x(x e. B -> x e. (C \ A))))
17		dfss2 3263	. 2 |- (A C_ (C \ B) <-> A.x(x e. A -> x e. (C \ B)))
18		dfss2 3263	. 2 |- (B C_ (C \ A) <-> A.x(x e. B -> x e. (C \ A)))
19	16, 17, 18	3bitr4g 279	1 |- ((A C_ C /\ B C_ C) -> (A C_ (C \ B) <-> B C_ (C \ A)))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   e. wcel 1710   \ cdif 3207   C_ wss 3258

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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260

This theorem is referenced by:  pssdifcom1  3636  pssdifcom2  3637