Theorem ssn0 3400

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 3399	. . . 4 |- ( ( A C_ B /\ B = (/) ) -> A = (/) )
2	1	ex 114	. . 3 |- ( A C_ B -> ( B = (/) -> A = (/) ) )
3	2	necon3d 2350	. 2 |- ( A C_ B -> ( A =/= (/) -> B =/= (/) ) )
4	3	imp 123	1 |- ( ( A C_ B /\ A =/= (/) ) -> B =/= (/) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   =/= wne 2306   C_ wss 3066   (/)c0 3358

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079  df-nul 3359

This theorem is referenced by: (None)