Theorem sseq0 3554

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 3266	. . 3 |- ( B = (/) -> ( A C_ B <-> A C_ (/) ) )
2		ss0 3553	. . 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 1398   C_ wss 3214   (/)c0 3512

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-dif 3216  df-in 3220  df-ss 3227  df-nul 3513

This theorem is referenced by:  ssn0  3555  ssdifin0  3595  fieq0  7276  fisumss  12103  strleund  13400  strleun  13401