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Theorem psstr 3673
 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 3664 . . 3 (𝐴𝐵𝐴𝐵)
2 pssss 3664 . . 3 (𝐵𝐶𝐵𝐶)
31, 2sylan9ss 3581 . 2 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
4 pssn2lp 3670 . . . 4 ¬ (𝐶𝐵𝐵𝐶)
5 psseq1 3656 . . . . 5 (𝐴 = 𝐶 → (𝐴𝐵𝐶𝐵))
65anbi1d 737 . . . 4 (𝐴 = 𝐶 → ((𝐴𝐵𝐵𝐶) ↔ (𝐶𝐵𝐵𝐶)))
74, 6mtbiri 316 . . 3 (𝐴 = 𝐶 → ¬ (𝐴𝐵𝐵𝐶))
87con2i 133 . 2 ((𝐴𝐵𝐵𝐶) → ¬ 𝐴 = 𝐶)
9 dfpss2 3654 . 2 (𝐴𝐶 ↔ (𝐴𝐶 ∧ ¬ 𝐴 = 𝐶))
103, 8, 9sylanbrc 695 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ⊆ wss 3540   ⊊ wpss 3541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ne 2782  df-in 3547  df-ss 3554  df-pss 3556 This theorem is referenced by:  sspsstr  3674  psssstr  3675  psstrd  3676  porpss  6839  inf3lem5  8412  ltsopr  9733
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