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Theorem sspsstr 3862
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 3856 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
2 psstr 3861 . . . . 5 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
32ex 397 . . . 4 (𝐴𝐵 → (𝐵𝐶𝐴𝐶))
4 psseq1 3844 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
54biimprd 238 . . . 4 (𝐴 = 𝐵 → (𝐵𝐶𝐴𝐶))
63, 5jaoi 846 . . 3 ((𝐴𝐵𝐴 = 𝐵) → (𝐵𝐶𝐴𝐶))
76imp 393 . 2 (((𝐴𝐵𝐴 = 𝐵) ∧ 𝐵𝐶) → 𝐴𝐶)
81, 7sylanb 570 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wo 836   = 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 837  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:  sspsstrd  3865  ordtr2  5910  php  8304  canthp1lem2  9681  suplem1pr  10080  fbfinnfr  21865  ppiltx  25124
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