Theorem sseq0 3533

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 3248	. . 3 |- ( B = (/) -> ( A C_ B <-> A C_ (/) ) )
2		ss0 3532	. . 3 |- ( A C_ (/) -> A = (/) )
3	1, 2	biimtrdi 163	. 2 |- ( B = (/) -> ( A C_ B -> A = (/) ) )
4	3	impcom 125	1 |- ( ( A C_ B /\ B = (/) ) -> A = (/) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   C_ wss 3197   (/)c0 3491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492

This theorem is referenced by:  ssn0  3534  ssdifin0  3573  fieq0  7139  fisumss  11898  strleund  13131  strleun  13132