Theorem ssdisj 3471

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 3454	. . . 4 |- ( ( B i^i C ) C_ (/) <-> ( B i^i C ) = (/) )
2		ssrin 3352	. . . . 5 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )
3		sstr2 3154	. . . . 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 3455	. 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 1348   i^i cin 3120   C_ wss 3121   (/)c0 3414

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415

This theorem is referenced by:  djudisj  5038  fimacnvdisj  5382  unfiin  6903  hashunlem  10739