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Theorem sseq0 3374
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 3091 . . 3  |-  ( B  =  (/)  ->  ( A 
C_  B  <->  A  C_  (/) ) )
2 ss0 3373 . . 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 1316    C_ wss 3041   (/)c0 3333
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054  df-nul 3334
This theorem is referenced by:  ssn0  3375  ssdifin0  3414  fieq0  6832  fisumss  11129  strleund  11974  strleun  11975
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