Theorem difin0ss 3617

Description: Difference, intersection, and subclass relationship. (Contributed by NM, 30-Apr-1994.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)

Assertion

Ref	Expression
difin0ss	|- (((A \ B) i^i C) = (/) -> (C C_ A -> C C_ B))

Proof of Theorem difin0ss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eq0 3565	. 2 |- (((A \ B) i^i C) = (/) <-> A.x -. x e. ((A \ B) i^i C))
2		iman 413	. . . . . 6 |- ((x e. C -> (x e. A -> x e. B)) <-> -. (x e. C /\ -. (x e. A -> x e. B)))
3		elin 3220	. . . . . . . 8 |- (x e. ((A \ B) i^i C) <-> (x e. (A \ B) /\ x e. C))
4		eldif 3222	. . . . . . . . 9 |- (x e. (A \ B) <-> (x e. A /\ -. x e. B))
5	4	anbi1i 676	. . . . . . . 8 |- ((x e. (A \ B) /\ x e. C) <-> ((x e. A /\ -. x e. B) /\ x e. C))
6	3, 5	bitri 240	. . . . . . 7 |- (x e. ((A \ B) i^i C) <-> ((x e. A /\ -. x e. B) /\ x e. C))
7		ancom 437	. . . . . . 7 |- ((x e. C /\ (x e. A /\ -. x e. B)) <-> ((x e. A /\ -. x e. B) /\ x e. C))
8		annim 414	. . . . . . . 8 |- ((x e. A /\ -. x e. B) <-> -. (x e. A -> x e. B))
9	8	anbi2i 675	. . . . . . 7 |- ((x e. C /\ (x e. A /\ -. x e. B)) <-> (x e. C /\ -. (x e. A -> x e. B)))
10	6, 7, 9	3bitr2i 264	. . . . . 6 |- (x e. ((A \ B) i^i C) <-> (x e. C /\ -. (x e. A -> x e. B)))
11	2, 10	xchbinxr 302	. . . . 5 |- ((x e. C -> (x e. A -> x e. B)) <-> -. x e. ((A \ B) i^i C))
12		ax-2 7	. . . . 5 |- ((x e. C -> (x e. A -> x e. B)) -> ((x e. C -> x e. A) -> (x e. C -> x e. B)))
13	11, 12	sylbir 204	. . . 4 |- (-. x e. ((A \ B) i^i C) -> ((x e. C -> x e. A) -> (x e. C -> x e. B)))
14	13	al2imi 1561	. . 3 |- (A.x -. x e. ((A \ B) i^i C) -> (A.x(x e. C -> x e. A) -> A.x(x e. C -> x e. B)))
15		dfss2 3263	. . 3 |- (C C_ A <-> A.x(x e. C -> x e. A))
16		dfss2 3263	. . 3 |- (C C_ B <-> A.x(x e. C -> x e. B))
17	14, 15, 16	3imtr4g 261	. 2 |- (A.x -. x e. ((A \ B) i^i C) -> (C C_ A -> C C_ B))
18	1, 17	sylbi 187	1 |- (((A \ B) i^i C) = (/) -> (C C_ A -> C C_ B))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710   \ cdif 3207   i^i cin 3209   C_ wss 3258  (/)c0 3551

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-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by: (None)