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Theorem sspsstrd 3598
Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3595. (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 3595 . 2  |-  ( ( A  C_  B  /\  B  C.  C )  ->  A  C.  C )
41, 2, 3syl2anc 659 1  |-  ( ph  ->  A  C.  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    C_ wss 3461    C. wpss 3462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-ne 2651  df-in 3468  df-ss 3475  df-pss 3477
This theorem is referenced by:  marypha1lem  7885  ackbij1lem15  8605  fin23lem38  8720  ltexprlem2  9404  mrieqv2d  15128
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