MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sspsstrd Unicode version

Theorem sspsstrd 3284
Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3281. (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 3281 . 2  |-  ( ( A  C_  B  /\  B  C.  C )  ->  A  C.  C )
41, 2, 3syl2anc 642 1  |-  ( ph  ->  A  C.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    C_ wss 3152    C. wpss 3153
This theorem is referenced by:  mrieqv2d  13541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ne 2448  df-in 3159  df-ss 3166  df-pss 3168
  Copyright terms: Public domain W3C validator