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Theorem sseq0 3435
Description: A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
sseq0  |-  ( ( A  C_  B  /\  B  =  (/) )  ->  A  =  (/) )

Proof of Theorem sseq0
StepHypRef Expression
1 sseq2 3152 . . 3  |-  ( B  =  (/)  ->  ( A 
C_  B  <->  A  C_  (/) ) )
2 ss0 3434 . . 3  |-  ( A 
C_  (/)  ->  A  =  (/) )
31, 2syl6bi 162 . 2  |-  ( B  =  (/)  ->  ( A 
C_  B  ->  A  =  (/) ) )
43impcom 124 1  |-  ( ( A  C_  B  /\  B  =  (/) )  ->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    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-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395
This theorem is referenced by:  ssn0  3436  ssdifin0  3475  fieq0  6920  fisumss  11289  strleund  12278  strleun  12279
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