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Theorem ssn0 3436
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
StepHypRef Expression
1 sseq0 3435 . . . 4  |-  ( ( A  C_  B  /\  B  =  (/) )  ->  A  =  (/) )
21ex 114 . . 3  |-  ( A 
C_  B  ->  ( B  =  (/)  ->  A  =  (/) ) )
32necon3d 2371 . 2  |-  ( A 
C_  B  ->  ( A  =/=  (/)  ->  B  =/=  (/) ) )
43imp 123 1  |-  ( ( A  C_  B  /\  A  =/=  (/) )  ->  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    =/= wne 2327    C_ wss 3102   (/)c0 3394
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395
This theorem is referenced by: (None)
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