Theorem ssdifin0 3496

Description: A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)

Assertion

Ref	Expression
ssdifin0	|- ( A C_ ( B \ C ) -> ( A i^i C ) = (/) )

Proof of Theorem ssdifin0

Step	Hyp	Ref	Expression
1		ssrin 3352	. 2 |- ( A C_ ( B \ C ) -> ( A i^i C ) C_ ( ( B \ C ) i^i C ) )
2		incom 3319	. . 3 |- ( ( B \ C ) i^i C ) = ( C i^i ( B \ C ) )
3		disjdif 3487	. . 3 |- ( C i^i ( B \ C ) ) = (/)
4	2, 3	eqtri 2191	. 2 |- ( ( B \ C ) i^i C ) = (/)
5		sseq0 3456	. 2 |- ( ( ( A i^i C ) C_ ( ( B \ C ) i^i C ) /\ ( ( B \ C ) i^i C ) = (/) ) -> ( A i^i C ) = (/) )
6	1, 4, 5	sylancl 411	1 |- ( A C_ ( B \ C ) -> ( A i^i C ) = (/) )

Syntax hints:   -> wi 4   = wceq 1348   \ cdif 3118   i^i cin 3120   C_ wss 3121   (/)c0 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415

This theorem is referenced by:  ssdifeq0  3497