Theorem ssdisj 3837

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 3776	. . . 4 |- ( ( B i^i C ) C_ (/) <-> ( B i^i C ) = (/) )
2		ssrin 3684	. . . . 5 |- ( A C_ B -> ( A i^i C ) C_ ( B i^i C ) )
3		sstr2 3472	. . . . 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 3777	. 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 1370   i^i cin 3436   C_ wss 3437   (/)c0 3746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3747

This theorem is referenced by:  djudisj  5374  fimacnvdisj  5698  marypha1lem  7795  ackbij1lem16  8516  ackbij1lem18  8518  fin23lem20  8618  fin23lem30  8623  elcls3  18820  neindisj  18854  imadifxp  26091  diophren  29301