Theorem ssdisj 3600

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 3580	. . . 4 |- ((B i^i C) C_ (/) <-> (B i^i C) = (/))
2		ssrin 3480	. . . . 5 |- (A C_ B -> (A i^i C) C_ (B i^i C))
3		sstr2 3279	. . . . 5 |- ((A i^i C) C_ (B i^i C) -> ((B i^i C) C_ (/) -> (A i^i C) C_ (/)))
4	2, 3	syl 15	. . . 4 |- (A C_ B -> ((B i^i C) C_ (/) -> (A i^i C) C_ (/)))
5	1, 4	syl5bir 209	. . 3 |- (A C_ B -> ((B i^i C) = (/) -> (A i^i C) C_ (/)))
6	5	imp 418	. 2 |- ((A C_ B /\ (B i^i C) = (/)) -> (A i^i C) C_ (/))
7		ss0 3581	. 2 |- ((A i^i C) C_ (/) -> (A i^i C) = (/))
8	6, 7	syl 15	1 |- ((A C_ B /\ (B i^i C) = (/)) -> (A i^i C) = (/))

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   i^i cin 3208   C_ wss 3257  (/)c0 3550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551

This theorem is referenced by:  fimacnvdisj  5244