Theorem sseq0 3456

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 3171	. . 3 |- ( B = (/) -> ( A C_ B <-> A C_ (/) ) )
2		ss0 3455	. . 3 |- ( A C_ (/) -> A = (/) )
3	1, 2	syl6bi 162	. 2 |- ( B = (/) -> ( A C_ B -> A = (/) ) )
4	3	impcom 124	1 |- ( ( A C_ B /\ B = (/) ) -> A = (/) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   C_ wss 3121   (/)c0 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415

This theorem is referenced by:  ssn0  3457  ssdifin0  3496  fieq0  6953  fisumss  11355  strleund  12506  strleun  12507