Theorem ssdifin0 3606

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 3456	. 2 |- ( A C_ ( B \ C ) -> ( A i^i C ) C_ ( ( B \ C ) i^i C ) )
2		incom 3421	. . 3 |- ( ( B \ C ) i^i C ) = ( C i^i ( B \ C ) )
3		disjdif 3596	. . 3 |- ( C i^i ( B \ C ) ) = (/)
4	2, 3	eqtri 2259	. 2 |- ( ( B \ C ) i^i C ) = (/)
5		sseq0 3564	. 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 417	1 |- ( A C_ ( B \ C ) -> ( A i^i C ) = (/) )

Syntax hints:   -> wi 4   = wceq 1402   \ cdif 3217   i^i cin 3219   C_ wss 3220   (/)c0 3520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521

This theorem is referenced by:  ssdifeq0  3607