Theorem ssdisj 3465

Description: Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)

Assertion

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

Proof of Theorem ssdisj

Step	Hyp	Ref	Expression
1		ss0b 3448	. . . 4 |- ( ( B i^i C ) C_ (/) <-> ( B i^i C ) = (/) )
2		ssrin 3347	. . . . 5 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )
3		sstr2 3149	. . . . 5 |- ( ( A i^i C ) C_ ( B i^i C ) -> ( ( B i^i C ) C_ (/) -> ( A i^i C ) C_ (/) ) )
4	2, 3	syl 14	. . . 4 |- ( A C_ B -> ( ( B i^i C ) C_ (/) -> ( A i^i C ) C_ (/) ) )
5	1, 4	syl5bir 152	. . 3 |- ( A C_ B -> ( ( B i^i C ) = (/) -> ( A i^i C ) C_ (/) ) )
6	5	imp 123	. 2 |- ( ( A C_ B /\ ( B i^i C ) = (/) ) -> ( A i^i C ) C_ (/) )
7		ss0 3449	. 2 |- ( ( A i^i C ) C_ (/) -> ( A i^i C ) = (/) )
8	6, 7	syl 14	1 |- ( ( A C_ B /\ ( B i^i C ) = (/) ) -> ( A i^i C ) = (/) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   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:  djudisj  5031  fimacnvdisj  5372  unfiin  6891  hashunlem  10717