Theorem ssdifin0 3304

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 3162	. 2 |- ( A C_ ( B \ C ) -> ( A i^i C ) C_ ( ( B \ C ) i^i C ) )
2		incom 3129	. . 3 |- ( ( B \ C ) i^i C ) = ( C i^i ( B \ C ) )
3		disjdif 3296	. . 3 |- ( C i^i ( B \ C ) ) = (/)
4	2, 3	eqtri 2060	. 2 |- ( ( B \ C ) i^i C ) = (/)
5		sseq0 3258	. 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 392	1 |- ( A C_ ( B \ C ) -> ( A i^i C ) = (/) )

Syntax hints:   -> wi 4   = wceq 1243   \ cdif 2914   i^i cin 2916   C_ wss 2917   (/)c0 3224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225

This theorem is referenced by:  ssdifeq0  3305