Theorem ssn0 3286

Description: A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)

Assertion

Ref	Expression
ssn0	|- ( ( A C_ B /\ A =/= (/) ) -> B =/= (/) )

Proof of Theorem ssn0

Step	Hyp	Ref	Expression
1		sseq0 3285	. . . 4 |- ( ( A C_ B /\ B = (/) ) -> A = (/) )
2	1	ex 113	. . 3 |- ( A C_ B -> ( B = (/) -> A = (/) ) )
3	2	necon3d 2289	. 2 |- ( A C_ B -> ( A =/= (/) -> B =/= (/) ) )
4	3	imp 122	1 |- ( ( A C_ B /\ A =/= (/) ) -> B =/= (/) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   =/= wne 2245   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-ne 2246  df-v 2603  df-dif 2975  df-in 2979  df-ss 2986  df-nul 3252

This theorem is referenced by: (None)