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

Theorem sspsstr 3996
Description: Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
Assertion
Ref Expression
sspsstr ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Proof of Theorem sspsstr
StepHypRef Expression
1 sspss 3990 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
2 psstr 3995 . . . . 5 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
32ex 416 . . . 4 (𝐴𝐵 → (𝐵𝐶𝐴𝐶))
4 psseq1 3978 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
54biimprd 251 . . . 4 (𝐴 = 𝐵 → (𝐵𝐶𝐴𝐶))
63, 5jaoi 856 . . 3 ((𝐴𝐵𝐴 = 𝐵) → (𝐵𝐶𝐴𝐶))
76imp 410 . 2 (((𝐴𝐵𝐴 = 𝐵) ∧ 𝐵𝐶) → 𝐴𝐶)
81, 7sylanb 584 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 846   = wceq 1542  wss 3843  wpss 3844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-v 3400  df-in 3850  df-ss 3860  df-pss 3862
This theorem is referenced by:  sspsstrd  3999  ordtr2  6216  php  8751  canthp1lem2  10153  suplem1pr  10552  fbfinnfr  22592  ppiltx  25914
  Copyright terms: Public domain W3C validator