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Theorem ssn0 3293
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 3292 . . . 4  |-  ( ( A  C_  B  /\  B  =  (/) )  ->  A  =  (/) )
21ex 113 . . 3  |-  ( A 
C_  B  ->  ( B  =  (/)  ->  A  =  (/) ) )
32necon3d 2290 . 2  |-  ( A 
C_  B  ->  ( A  =/=  (/)  ->  B  =/=  (/) ) )
43imp 122 1  |-  ( ( A  C_  B  /\  A  =/=  (/) )  ->  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    =/= wne 2246    C_ wss 2974   (/)c0 3258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-v 2604  df-dif 2976  df-in 2980  df-ss 2987  df-nul 3259
This theorem is referenced by: (None)
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