Theorem ssconb 2738

Description: Contraposition law for subsets.

Assertion

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

Proof of Theorem ssconb

Step	Hyp	Ref	Expression
1		pm5.1 740	. . . . . . 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)))
2		ssel 2615	. . . . . . 7 |- (A C_ C -> (x e. A -> x e. C))
3		ssel 2615	. . . . . . 7 |- (B C_ C -> (x e. B -> x e. C))
4	1, 2, 3	syl2an 503	. . . . . 6 |- ((A C_ C /\ B C_ C) -> ((x e. A -> x e. C) <-> (x e. B -> x e. C)))
5		con2b 182	. . . . . . 7 |- ((x e. A -> -. x e. B) <-> (x e. B -> -. x e. A))
6	5	a1i 8	. . . . . 6 |- ((A C_ C /\ B C_ C) -> ((x e. A -> -. x e. B) <-> (x e. B -> -. x e. A)))
7	4, 6	anbi12d 690	. . . . 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 659	. . . . 5 |- ((x e. A -> (x e. C /\ -. x e. B)) <-> ((x e. A -> x e. C) /\ (x e. A -> -. x e. B)))
9		jcab 659	. . . . 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 614	. . . 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 2609	. . . . 5 |- (x e. (C \ B) <-> (x e. C /\ -. x e. B))
12	11	imbi2i 202	. . . 4 |- ((x e. A -> x e. (C \ B)) <-> (x e. A -> (x e. C /\ -. x e. B)))
13		eldif 2609	. . . . 5 |- (x e. (C \ A) <-> (x e. C /\ -. x e. A))
14	13	imbi2i 202	. . . 4 |- ((x e. B -> x e. (C \ A)) <-> (x e. B -> (x e. C /\ -. x e. A)))
15	10, 12, 14	3bitr4g 614	. . 3 |- ((A C_ C /\ B C_ C) -> ((x e. A -> x e. (C \ B)) <-> (x e. B -> x e. (C \ A))))
16	15	albidv 1656	. 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 2610	. 2 |- (A C_ (C \ B) <-> A.x(x e. A -> x e. (C \ B)))
18		dfss2 2610	. 2 |- (B C_ (C \ A) <-> A.x(x e. B -> x e. (C \ A)))
19	16, 17, 18	3bitr4g 614	1 |- ((A C_ C /\ B C_ C) -> (A C_ (C \ B) <-> B C_ (C \ A)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   e. wcel 1300   \ cdif 2590   C_ wss 2593

This theorem is referenced by:  sbthlem1 5510  sbthlem2 5511  clsval2 8961  conss2 16420

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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