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Theorem syl5sseq 3057
Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Hypotheses
Ref Expression
syl5sseq.1  |-  B  C_  A
syl5sseq.2  |-  ( ph  ->  A  =  C )
Assertion
Ref Expression
syl5sseq  |-  ( ph  ->  B  C_  C )

Proof of Theorem syl5sseq
StepHypRef Expression
1 syl5sseq.2 . 2  |-  ( ph  ->  A  =  C )
2 syl5sseq.1 . 2  |-  B  C_  A
3 sseq2 3031 . . 3  |-  ( A  =  C  ->  ( B  C_  A  <->  B  C_  C
) )
43biimpa 290 . 2  |-  ( ( A  =  C  /\  B  C_  A )  ->  B  C_  C )
51, 2, 4sylancl 404 1  |-  ( ph  ->  B  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    C_ wss 2983
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 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-in 2989  df-ss 2996
This theorem is referenced by:  fndmdif  5325  fneqeql2  5329  fconst4m  5434  f1opw2  5758  ecss  6235  fopwdom  6402  phplem2  6410  nn0supp  8443  monoord2  9588
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