Theorem sseq0 3285

Description: A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
sseq0	|- ( ( A C_ B /\ B = (/) ) -> A = (/) )

Proof of Theorem sseq0

Step	Hyp	Ref	Expression
1		sseq2 3021	. . 3 |- ( B = (/) -> ( A C_ B <-> A C_ (/) ) )
2		ss0 3284	. . 3 |- ( A C_ (/) -> A = (/) )
3	1, 2	syl6bi 161	. 2 |- ( B = (/) -> ( A C_ B -> A = (/) ) )
4	3	impcom 123	1 |- ( ( A C_ B /\ B = (/) ) -> A = (/) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   C_ wss 2973   (/)c0 3251

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-in 2979  df-ss 2986  df-nul 3252

This theorem is referenced by:  ssn0  3286  ssdifin0  3324