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

Theorem psssstr 3863
Description: Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
Assertion
Ref Expression
psssstr ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Proof of Theorem psssstr
StepHypRef Expression
1 sspss 3856 . 2 (𝐵𝐶 ↔ (𝐵𝐶𝐵 = 𝐶))
2 psstr 3861 . . . . 5 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
32ex 397 . . . 4 (𝐴𝐵 → (𝐵𝐶𝐴𝐶))
4 psseq2 3845 . . . . 5 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
54biimpcd 239 . . . 4 (𝐴𝐵 → (𝐵 = 𝐶𝐴𝐶))
63, 5jaod 838 . . 3 (𝐴𝐵 → ((𝐵𝐶𝐵 = 𝐶) → 𝐴𝐶))
76imp 393 . 2 ((𝐴𝐵 ∧ (𝐵𝐶𝐵 = 𝐶)) → 𝐴𝐶)
81, 7sylan2b 573 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wo 826   = wceq 1631  wss 3723  wpss 3724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-ne 2944  df-in 3730  df-ss 3737  df-pss 3739
This theorem is referenced by:  psssstrd  3866  suplem1pr  10076  atexch  29580  bj-2upln0  33342  bj-2upln1upl  33343
  Copyright terms: Public domain W3C validator