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Theorem sspsstr 3690
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 3684 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
2 psstr 3689 . . . . 5 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
32ex 450 . . . 4 (𝐴𝐵 → (𝐵𝐶𝐴𝐶))
4 psseq1 3672 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
54biimprd 238 . . . 4 (𝐴 = 𝐵 → (𝐵𝐶𝐴𝐶))
63, 5jaoi 394 . . 3 ((𝐴𝐵𝐴 = 𝐵) → (𝐵𝐶𝐴𝐶))
76imp 445 . 2 (((𝐴𝐵𝐴 = 𝐵) ∧ 𝐵𝐶) → 𝐴𝐶)
81, 7sylanb 489 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384   = wceq 1480  wss 3555  wpss 3556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ne 2791  df-in 3562  df-ss 3569  df-pss 3571
This theorem is referenced by:  sspsstrd  3693  ordtr2  5727  php  8088  canthp1lem2  9419  suplem1pr  9818  fbfinnfr  21555  ppiltx  24803
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