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Theorem sspsstrd 3457
Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3454. (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 3454 . 2  |-  ( ( A  C_  B  /\  B  C.  C )  ->  A  C.  C )
41, 2, 3syl2anc 644 1  |-  ( ph  ->  A  C.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3322    C. wpss 3323
This theorem is referenced by:  marypha1lem  7440  ackbij1lem15  8116  fin23lem38  8231  ltexprlem2  8916  mrieqv2d  13866
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-ne 2603  df-in 3329  df-ss 3336  df-pss 3338
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