Theorem ssdisj 3581

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 3562	. . . 4 |- ( ( B i^i C ) C_ (/) <-> ( B i^i C ) = (/) )
2		ssrin 3456	. . . . 5 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )
3		sstr2 3255	. . . . 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	biimtrrid 153	. . 3 |- ( A C_ B -> ( ( B i^i C ) = (/) -> ( A i^i C ) C_ (/) ) )
6	5	imp 124	. 2 |- ( ( A C_ B /\ ( B i^i C ) = (/) ) -> ( A i^i C ) C_ (/) )
7		ss0 3563	. 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 104   = wceq 1402   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:  djudisj  5213  fimacnvdisj  5574  unfiin  7226  hashunlem  11225