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Theorem psstr 3853
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 3844 . . 3 (𝐴𝐵𝐴𝐵)
2 pssss 3844 . . 3 (𝐵𝐶𝐵𝐶)
31, 2sylan9ss 3757 . 2 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
4 pssn2lp 3850 . . . 4 ¬ (𝐶𝐵𝐵𝐶)
5 psseq1 3836 . . . . 5 (𝐴 = 𝐶 → (𝐴𝐵𝐶𝐵))
65anbi1d 743 . . . 4 (𝐴 = 𝐶 → ((𝐴𝐵𝐵𝐶) ↔ (𝐶𝐵𝐵𝐶)))
74, 6mtbiri 316 . . 3 (𝐴 = 𝐶 → ¬ (𝐴𝐵𝐵𝐶))
87con2i 134 . 2 ((𝐴𝐵𝐵𝐶) → ¬ 𝐴 = 𝐶)
9 dfpss2 3834 . 2 (𝐴𝐶 ↔ (𝐴𝐶 ∧ ¬ 𝐴 = 𝐶))
103, 8, 9sylanbrc 701 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wss 3715  wpss 3716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-ne 2933  df-in 3722  df-ss 3729  df-pss 3731
This theorem is referenced by:  sspsstr  3854  psssstr  3855  psstrd  3856  porpss  7107  inf3lem5  8704  ltsopr  10066
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