Theorem ssdisj 2318

Description: Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)

Assertion

Ref	Expression
ssdisj	|- ((A (_ B /\ (B i^i C) = (/)) -> (A i^i C) = (/))

Proof of Theorem ssdisj

Step	Hyp	Ref	Expression
1		ssrin 2234	. . . . 5 |- (A (_ B -> (A i^i C) (_ (B i^i C))
2		sstr2 2071	. . . . 5 |- ((A i^i C) (_ (B i^i C) -> ((B i^i C) (_ (/) -> (A i^i C) (_ (/)))
3	1, 2	syl 10	. . . 4 |- (A (_ B -> ((B i^i C) (_ (/) -> (A i^i C) (_ (/)))
4		ss0b 2302	. . . 4 |- ((B i^i C) (_ (/) <-> (B i^i C) = (/))
5	3, 4	syl5ibr 207	. . 3 |- (A (_ B -> ((B i^i C) = (/) -> (A i^i C) (_ (/)))
6	5	imp 350	. 2 |- ((A (_ B /\ (B i^i C) = (/)) -> (A i^i C) (_ (/))
7		ss0 2303	. 2 |- ((A i^i C) (_ (/) -> (A i^i C) = (/))
8	6, 7	syl 10	1 |- ((A (_ B /\ (B i^i C) = (/)) -> (A i^i C) = (/))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   i^i cin 2046   (_ wss 2047  (/)c0 2280

This theorem is referenced by:  fimacnvdisj 3649  elcls3 7711  neindisj 7731

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-nul 2281

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