Theorem ssdisj 3876

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 3815	. . . 4 |- ( ( B i^i C ) C_ (/) <-> ( B i^i C ) = (/) )
2		ssrin 3723	. . . . 5 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )
3		sstr2 3511	. . . . 5 |- ( ( A i^i C ) C_ ( B i^i C ) -> ( ( B i^i C ) C_ (/) -> ( A i^i C ) C_ (/) ) )
4	2, 3	syl 16	. . . 4 |- ( A C_ B -> ( ( B i^i C ) C_ (/) -> ( A i^i C ) C_ (/) ) )
5	1, 4	syl5bir 218	. . 3 |- ( A C_ B -> ( ( B i^i C ) = (/) -> ( A i^i C ) C_ (/) ) )
6	5	imp 429	. 2 |- ( ( A C_ B /\ ( B i^i C ) = (/) ) -> ( A i^i C ) C_ (/) )
7		ss0 3816	. 2 |- ( ( A i^i C ) C_ (/) -> ( A i^i C ) = (/) )
8	6, 7	syl 16	1 |- ( ( A C_ B /\ ( B i^i C ) = (/) ) -> ( A i^i C ) = (/) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   i^i cin 3475   C_ wss 3476   (/)c0 3785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786

This theorem is referenced by:  djudisj  5432  fimacnvdisj  5761  marypha1lem  7889  ackbij1lem16  8611  ackbij1lem18  8613  fin23lem20  8713  fin23lem30  8718  elcls3  19347  neindisj  19381  imadifxp  27128  diophren  30349