Theorem difin0ss 3946

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 3929	. 2 |- ( ( ( A \ B ) i^i C ) = (/) <-> A. x -. x e. ( ( A \ B ) i^i C ) )
2		iman 440	. . . . . 6 |- ( ( x e. C -> ( x e. A -> x e. B ) ) <-> -. ( x e. C /\ -. ( x e. A -> x e. B ) ) )
3		elin 3796	. . . . . . 7 |- ( x e. ( ( A \ B ) i^i C ) <-> ( x e. ( A \ B ) /\ x e. C ) )
4		eldif 3584	. . . . . . . 8 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
5	4	anbi2ci 732	. . . . . . 7 |- ( ( x e. ( A \ B ) /\ x e. C ) <-> ( x e. C /\ ( x e. A /\ -. x e. B ) ) )
6		annim 441	. . . . . . . 8 |- ( ( x e. A /\ -. x e. B ) <-> -. ( x e. A -> x e. B ) )
7	6	anbi2i 730	. . . . . . 7 |- ( ( x e. C /\ ( x e. A /\ -. x e. B ) ) <-> ( x e. C /\ -. ( x e. A -> x e. B ) ) )
8	3, 5, 7	3bitri 286	. . . . . 6 |- ( x e. ( ( A \ B ) i^i C ) <-> ( x e. C /\ -. ( x e. A -> x e. B ) ) )
9	2, 8	xchbinxr 325	. . . . 5 |- ( ( x e. C -> ( x e. A -> x e. B ) ) <-> -. x e. ( ( A \ B ) i^i C ) )
10		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 ) ) )
11	9, 10	sylbir 225	. . . 4 |- ( -. x e. ( ( A \ B ) i^i C ) -> ( ( x e. C -> x e. A ) -> ( x e. C -> x e. B ) ) )
12	11	al2imi 1743	. . 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 ) ) )
13		dfss2 3591	. . 3 |- ( C C_ A <-> A. x ( x e. C -> x e. A ) )
14		dfss2 3591	. . 3 |- ( C C_ B <-> A. x ( x e. C -> x e. B ) )
15	12, 13, 14	3imtr4g 285	. 2 |- ( A. x -. x e. ( ( A \ B ) i^i C ) -> ( C C_ A -> C C_ B ) )
16	1, 15	sylbi 207	1 |- ( ( ( A \ B ) i^i C ) = (/) -> ( C C_ A -> C C_ B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  tz7.7  5749  tfi  7053  lebnumlem3  22762