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Theorem sspsstrd 3565
Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3562. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
sspsstrd.1  |-  ( ph  ->  A  C_  B )
sspsstrd.2  |-  ( ph  ->  B  C.  C )
Assertion
Ref Expression
sspsstrd  |-  ( ph  ->  A  C.  C )

Proof of Theorem sspsstrd
StepHypRef Expression
1 sspsstrd.1 . 2  |-  ( ph  ->  A  C_  B )
2 sspsstrd.2 . 2  |-  ( ph  ->  B  C.  C )
3 sspsstr 3562 . 2  |-  ( ( A  C_  B  /\  B  C.  C )  ->  A  C.  C )
41, 2, 3syl2anc 661 1  |-  ( ph  ->  A  C.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    C_ wss 3429    C. wpss 3430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-ne 2646  df-in 3436  df-ss 3443  df-pss 3445
This theorem is referenced by:  marypha1lem  7787  ackbij1lem15  8507  fin23lem38  8622  ltexprlem2  9310  mrieqv2d  14688
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