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Theorem sseq0 3456
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 3171 . . 3  |-  ( B  =  (/)  ->  ( A 
C_  B  <->  A  C_  (/) ) )
2 ss0 3455 . . 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 1348    C_ wss 3121   (/)c0 3414
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415
This theorem is referenced by:  ssn0  3457  ssdifin0  3496  fieq0  6953  fisumss  11355  strleund  12506  strleun  12507
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