MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psssstrd Structured version   Visualization version   GIF version

Theorem psssstrd 3864
Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 3861. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
psssstrd.1 (𝜑𝐴𝐵)
psssstrd.2 (𝜑𝐵𝐶)
Assertion
Ref Expression
psssstrd (𝜑𝐴𝐶)

Proof of Theorem psssstrd
StepHypRef Expression
1 psssstrd.1 . 2 (𝜑𝐴𝐵)
2 psssstrd.2 . 2 (𝜑𝐵𝐶)
3 psssstr 3861 . 2 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
41, 2, 3syl2anc 565 1 (𝜑𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3721  wpss 3722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-ne 2943  df-in 3728  df-ss 3735  df-pss 3737
This theorem is referenced by:  ackbij1lem15  9257  lsatssn0  34804  lsatexch  34845  lsatcvatlem  34851  lkrpssN  34965
  Copyright terms: Public domain W3C validator