Theorem difin0ss 3833

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 3747	. 2 |- ( ( ( A \ B ) i^i C ) = (/) <-> A. x -. x e. ( ( A \ B ) i^i C ) )
2		iman 426	. . . . . 6 |- ( ( x e. C -> ( x e. A -> x e. B ) ) <-> -. ( x e. C /\ -. ( x e. A -> x e. B ) ) )
3		elin 3617	. . . . . . . 8 |- ( x e. ( ( A \ B ) i^i C ) <-> ( x e. ( A \ B ) /\ x e. C ) )
4		eldif 3414	. . . . . . . . 9 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
5	4	anbi1i 701	. . . . . . . 8 |- ( ( x e. ( A \ B ) /\ x e. C ) <-> ( ( x e. A /\ -. x e. B ) /\ x e. C ) )
6	3, 5	bitri 253	. . . . . . 7 |- ( x e. ( ( A \ B ) i^i C ) <-> ( ( x e. A /\ -. x e. B ) /\ x e. C ) )
7		ancom 452	. . . . . . 7 |- ( ( x e. C /\ ( x e. A /\ -. x e. B ) ) <-> ( ( x e. A /\ -. x e. B ) /\ x e. C ) )
8		annim 427	. . . . . . . 8 |- ( ( x e. A /\ -. x e. B ) <-> -. ( x e. A -> x e. B ) )
9	8	anbi2i 700	. . . . . . 7 |- ( ( x e. C /\ ( x e. A /\ -. x e. B ) ) <-> ( x e. C /\ -. ( x e. A -> x e. B ) ) )
10	6, 7, 9	3bitr2i 277	. . . . . 6 |- ( x e. ( ( A \ B ) i^i C ) <-> ( x e. C /\ -. ( x e. A -> x e. B ) ) )
11	2, 10	xchbinxr 313	. . . . 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 217	. . . 4 |- ( -. x e. ( ( A \ B ) i^i C ) -> ( ( x e. C -> x e. A ) -> ( x e. C -> x e. B ) ) )
14	13	al2imi 1687	. . 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 3421	. . 3 |- ( C C_ A <-> A. x ( x e. C -> x e. A ) )
16		dfss2 3421	. . 3 |- ( C C_ B <-> A. x ( x e. C -> x e. B ) )
17	14, 15, 16	3imtr4g 274	. 2 |- ( A. x -. x e. ( ( A \ B ) i^i C ) -> ( C C_ A -> C C_ B ) )
18	1, 17	sylbi 199	1 |- ( ( ( A \ B ) i^i C ) = (/) -> ( C C_ A -> C C_ B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   A.wal 1442   = wceq 1444   e. wcel 1887   \ cdif 3401   i^i cin 3403   C_ wss 3404   (/)c0 3731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-v 3047  df-dif 3407  df-in 3411  df-ss 3418  df-nul 3732

This theorem is referenced by:  tz7.7  5449  tfi  6680  lebnumlem3  21991  lebnumlem3OLD  21994