Theorem ssdifin0 3449

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 3306	. 2 |- ( A C_ ( B \ C ) -> ( A i^i C ) C_ ( ( B \ C ) i^i C ) )
2		incom 3273	. . 3 |- ( ( B \ C ) i^i C ) = ( C i^i ( B \ C ) )
3		disjdif 3440	. . 3 |- ( C i^i ( B \ C ) ) = (/)
4	2, 3	eqtri 2161	. 2 |- ( ( B \ C ) i^i C ) = (/)
5		sseq0 3409	. 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 1332   \ cdif 3073   i^i cin 3075   C_ wss 3076   (/)c0 3368

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089  df-nul 3369

This theorem is referenced by:  ssdifeq0  3450