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Theorem syl5ss 3011
Description: Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
Hypotheses
Ref Expression
syl5ss.1  |-  A  C_  B
syl5ss.2  |-  ( ph  ->  B  C_  C )
Assertion
Ref Expression
syl5ss  |-  ( ph  ->  A  C_  C )

Proof of Theorem syl5ss
StepHypRef Expression
1 syl5ss.1 . . 3  |-  A  C_  B
21a1i 9 . 2  |-  ( ph  ->  A  C_  B )
3 syl5ss.2 . 2  |-  ( ph  ->  B  C_  C )
42, 3sstrd 3010 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 2974
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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  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-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987
This theorem is referenced by:  fimacnv  5328  smores2  5943  f1imaen2g  6340  phplem4dom  6397  genipv  6761  fzossnn0  9261
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