Theorem ssn0 3566

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 3565	. . . 4 |- ( ( A C_ B /\ B = (/) ) -> A = (/) )
2	1	ex 115	. . 3 |- ( A C_ B -> ( B = (/) -> A = (/) ) )
3	2	necon3d 2464	. 2 |- ( A C_ B -> ( A =/= (/) -> B =/= (/) ) )
4	3	imp 124	1 |- ( ( A C_ B /\ A =/= (/) ) -> B =/= (/) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   =/= wne 2420   C_ wss 3220   (/)c0 3520

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521

This theorem is referenced by: (None)