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Theorem sspsstr 3909
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 3903 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
2 psstr 3908 . . . . 5 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
32ex 402 . . . 4 (𝐴𝐵 → (𝐵𝐶𝐴𝐶))
4 psseq1 3891 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
54biimprd 240 . . . 4 (𝐴 = 𝐵 → (𝐵𝐶𝐴𝐶))
63, 5jaoi 884 . . 3 ((𝐴𝐵𝐴 = 𝐵) → (𝐵𝐶𝐴𝐶))
76imp 396 . 2 (((𝐴𝐵𝐴 = 𝐵) ∧ 𝐵𝐶) → 𝐴𝐶)
81, 7sylanb 577 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wo 874   = wceq 1653  wss 3769  wpss 3770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-ne 2972  df-in 3776  df-ss 3783  df-pss 3785
This theorem is referenced by:  sspsstrd  3912  ordtr2  5985  php  8386  canthp1lem2  9763  suplem1pr  10162  fbfinnfr  21973  ppiltx  25255
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