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Theorem sspsstr 4085
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 4079 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
2 psstr 4084 . . . . 5 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
32ex 413 . . . 4 (𝐴𝐵 → (𝐵𝐶𝐴𝐶))
4 psseq1 4067 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
54biimprd 249 . . . 4 (𝐴 = 𝐵 → (𝐵𝐶𝐴𝐶))
63, 5jaoi 853 . . 3 ((𝐴𝐵𝐴 = 𝐵) → (𝐵𝐶𝐴𝐶))
76imp 407 . 2 (((𝐴𝐵𝐴 = 𝐵) ∧ 𝐵𝐶) → 𝐴𝐶)
81, 7sylanb 581 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 843   = wceq 1530  wss 3939  wpss 3940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-ne 3021  df-in 3946  df-ss 3955  df-pss 3957
This theorem is referenced by:  sspsstrd  4088  ordtr2  6232  php  8693  canthp1lem2  10067  suplem1pr  10466  fbfinnfr  22367  ppiltx  25670
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