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Theorem psstr 3672
Description: Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
Assertion
Ref Expression
psstr ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Proof of Theorem psstr
StepHypRef Expression
1 pssss 3663 . . 3 (𝐴𝐵𝐴𝐵)
2 pssss 3663 . . 3 (𝐵𝐶𝐵𝐶)
31, 2sylan9ss 3580 . 2 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
4 pssn2lp 3669 . . . 4 ¬ (𝐶𝐵𝐵𝐶)
5 psseq1 3655 . . . . 5 (𝐴 = 𝐶 → (𝐴𝐵𝐶𝐵))
65anbi1d 736 . . . 4 (𝐴 = 𝐶 → ((𝐴𝐵𝐵𝐶) ↔ (𝐶𝐵𝐵𝐶)))
74, 6mtbiri 315 . . 3 (𝐴 = 𝐶 → ¬ (𝐴𝐵𝐵𝐶))
87con2i 132 . 2 ((𝐴𝐵𝐵𝐶) → ¬ 𝐴 = 𝐶)
9 dfpss2 3653 . 2 (𝐴𝐶 ↔ (𝐴𝐶 ∧ ¬ 𝐴 = 𝐶))
103, 8, 9sylanbrc 694 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wss 3539  wpss 3540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-ne 2781  df-in 3546  df-ss 3553  df-pss 3555
This theorem is referenced by:  sspsstr  3673  psssstr  3674  psstrd  3675  porpss  6817  inf3lem5  8390  ltsopr  9711
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