Theorem ssdifin0 3490

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 3347	. 2 |- ( A C_ ( B \ C ) -> ( A i^i C ) C_ ( ( B \ C ) i^i C ) )
2		incom 3314	. . 3 |- ( ( B \ C ) i^i C ) = ( C i^i ( B \ C ) )
3		disjdif 3481	. . 3 |- ( C i^i ( B \ C ) ) = (/)
4	2, 3	eqtri 2186	. 2 |- ( ( B \ C ) i^i C ) = (/)
5		sseq0 3450	. 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 410	1 |- ( A C_ ( B \ C ) -> ( A i^i C ) = (/) )

Syntax hints:   -> wi 4   = wceq 1343   \ cdif 3113   i^i cin 3115   C_ wss 3116   (/)c0 3409

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410

This theorem is referenced by:  ssdifeq0  3491