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Theorem psstr 4065
 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 4056 . . 3 (𝐴𝐵𝐴𝐵)
2 pssss 4056 . . 3 (𝐵𝐶𝐵𝐶)
31, 2sylan9ss 3964 . 2 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
4 pssn2lp 4062 . . . 4 ¬ (𝐶𝐵𝐵𝐶)
5 psseq1 4048 . . . . 5 (𝐴 = 𝐶 → (𝐴𝐵𝐶𝐵))
65anbi1d 632 . . . 4 (𝐴 = 𝐶 → ((𝐴𝐵𝐵𝐶) ↔ (𝐶𝐵𝐵𝐶)))
74, 6mtbiri 330 . . 3 (𝐴 = 𝐶 → ¬ (𝐴𝐵𝐵𝐶))
87con2i 141 . 2 ((𝐴𝐵𝐵𝐶) → ¬ 𝐴 = 𝐶)
9 dfpss2 4046 . 2 (𝐴𝐶 ↔ (𝐴𝐶 ∧ ¬ 𝐴 = 𝐶))
103, 8, 9sylanbrc 586 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ⊆ wss 3918   ⊊ wpss 3919 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ne 3014  df-v 3481  df-in 3925  df-ss 3935  df-pss 3937 This theorem is referenced by:  sspsstr  4066  psssstr  4067  psstrd  4068  porpss  7438  inf3lem5  9081  ltsopr  10441
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