Theorem difin0ss 3897

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 3809	. 2 |- ( ( ( A \ B ) i^i C ) = (/) <-> A. x -. x e. ( ( A \ B ) i^i C ) )
2		iman 424	. . . . . 6 |- ( ( x e. C -> ( x e. A -> x e. B ) ) <-> -. ( x e. C /\ -. ( x e. A -> x e. B ) ) )
3		elin 3683	. . . . . . . 8 |- ( x e. ( ( A \ B ) i^i C ) <-> ( x e. ( A \ B ) /\ x e. C ) )
4		eldif 3481	. . . . . . . . 9 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
5	4	anbi1i 695	. . . . . . . 8 |- ( ( x e. ( A \ B ) /\ x e. C ) <-> ( ( x e. A /\ -. x e. B ) /\ x e. C ) )
6	3, 5	bitri 249	. . . . . . 7 |- ( x e. ( ( A \ B ) i^i C ) <-> ( ( x e. A /\ -. x e. B ) /\ x e. C ) )
7		ancom 450	. . . . . . 7 |- ( ( x e. C /\ ( x e. A /\ -. x e. B ) ) <-> ( ( x e. A /\ -. x e. B ) /\ x e. C ) )
8		annim 425	. . . . . . . 8 |- ( ( x e. A /\ -. x e. B ) <-> -. ( x e. A -> x e. B ) )
9	8	anbi2i 694	. . . . . . 7 |- ( ( x e. C /\ ( x e. A /\ -. x e. B ) ) <-> ( x e. C /\ -. ( x e. A -> x e. B ) ) )
10	6, 7, 9	3bitr2i 273	. . . . . 6 |- ( x e. ( ( A \ B ) i^i C ) <-> ( x e. C /\ -. ( x e. A -> x e. B ) ) )
11	2, 10	xchbinxr 311	. . . . 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 213	. . . 4 |- ( -. x e. ( ( A \ B ) i^i C ) -> ( ( x e. C -> x e. A ) -> ( x e. C -> x e. B ) ) )
14	13	al2imi 1637	. . 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 3488	. . 3 |- ( C C_ A <-> A. x ( x e. C -> x e. A ) )
16		dfss2 3488	. . 3 |- ( C C_ B <-> A. x ( x e. C -> x e. B ) )
17	14, 15, 16	3imtr4g 270	. 2 |- ( A. x -. x e. ( ( A \ B ) i^i C ) -> ( C C_ A -> C C_ B ) )
18	1, 17	sylbi 195	1 |- ( ( ( A \ B ) i^i C ) = (/) -> ( C C_ A -> C C_ B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   A.wal 1393   = wceq 1395   e. wcel 1819   \ cdif 3468   i^i cin 3470   C_ wss 3471   (/)c0 3793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3794

This theorem is referenced by:  tz7.7  4913  tfi  6687  lebnumlem3  21589