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Theorem sspsstr 4104
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 4098 . 2 (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
2 psstr 4103 . . . . 5 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
32ex 411 . . . 4 (𝐴𝐵 → (𝐵𝐶𝐴𝐶))
4 psseq1 4086 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
54biimprd 247 . . . 4 (𝐴 = 𝐵 → (𝐵𝐶𝐴𝐶))
63, 5jaoi 855 . . 3 ((𝐴𝐵𝐴 = 𝐵) → (𝐵𝐶𝐴𝐶))
76imp 405 . 2 (((𝐴𝐵𝐴 = 𝐵) ∧ 𝐵𝐶) → 𝐴𝐶)
81, 7sylanb 579 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wo 845   = wceq 1534  wss 3947  wpss 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1775  df-cleq 2718  df-ne 2931  df-ss 3964  df-pss 3967
This theorem is referenced by:  sspsstrd  4107  ordtr2  6420  php  9244  phpOLD  9256  canthp1lem2  10696  suplem1pr  11095  fbfinnfr  23836  ppiltx  27205
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