Theorem sseq0 3536

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 3251	. . 3 |- ( B = (/) -> ( A C_ B <-> A C_ (/) ) )
2		ss0 3535	. . 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 1397   C_ wss 3200   (/)c0 3494

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495

This theorem is referenced by:  ssn0  3537  ssdifin0  3576  fieq0  7174  fisumss  11952  strleund  13185  strleun  13186