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

Theorem psssstr 3675
 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 3668 . 2 (𝐵𝐶 ↔ (𝐵𝐶𝐵 = 𝐶))
2 psstr 3673 . . . . 5 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
32ex 449 . . . 4 (𝐴𝐵 → (𝐵𝐶𝐴𝐶))
4 psseq2 3657 . . . . 5 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
54biimpcd 238 . . . 4 (𝐴𝐵 → (𝐵 = 𝐶𝐴𝐶))
63, 5jaod 394 . . 3 (𝐴𝐵 → ((𝐵𝐶𝐵 = 𝐶) → 𝐴𝐶))
76imp 444 . 2 ((𝐴𝐵 ∧ (𝐵𝐶𝐵 = 𝐶)) → 𝐴𝐶)
81, 7sylan2b 491 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ⊆ wss 3540   ⊊ wpss 3541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ne 2782  df-in 3547  df-ss 3554  df-pss 3556 This theorem is referenced by:  psssstrd  3678  suplem1pr  9753  atexch  28624  bj-2upln0  32204  bj-2upln1upl  32205
 Copyright terms: Public domain W3C validator