Theorem ssn0 3451

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 3450	. . . 4 |- ( ( A C_ B /\ B = (/) ) -> A = (/) )
2	1	ex 114	. . 3 |- ( A C_ B -> ( B = (/) -> A = (/) ) )
3	2	necon3d 2380	. 2 |- ( A C_ B -> ( A =/= (/) -> B =/= (/) ) )
4	3	imp 123	1 |- ( ( A C_ B /\ A =/= (/) ) -> B =/= (/) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   =/= wne 2336   C_ wss 3116   (/)c0 3409

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410

This theorem is referenced by: (None)