Theorem ssn0 3511

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 3510	. . . 4 |- ( ( A C_ B /\ B = (/) ) -> A = (/) )
2	1	ex 115	. . 3 |- ( A C_ B -> ( B = (/) -> A = (/) ) )
3	2	necon3d 2422	. 2 |- ( A C_ B -> ( A =/= (/) -> B =/= (/) ) )
4	3	imp 124	1 |- ( ( A C_ B /\ A =/= (/) ) -> B =/= (/) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   =/= wne 2378   C_ wss 3174   (/)c0 3468

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187  df-nul 3469

This theorem is referenced by: (None)