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Theorem psstr 4078
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 4069 . . 3 (𝐴𝐵𝐴𝐵)
2 pssss 4069 . . 3 (𝐵𝐶𝐵𝐶)
31, 2sylan9ss 3977 . 2 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
4 pssn2lp 4075 . . . 4 ¬ (𝐶𝐵𝐵𝐶)
5 psseq1 4061 . . . . 5 (𝐴 = 𝐶 → (𝐴𝐵𝐶𝐵))
65anbi1d 629 . . . 4 (𝐴 = 𝐶 → ((𝐴𝐵𝐵𝐶) ↔ (𝐶𝐵𝐵𝐶)))
74, 6mtbiri 328 . . 3 (𝐴 = 𝐶 → ¬ (𝐴𝐵𝐵𝐶))
87con2i 141 . 2 ((𝐴𝐵𝐵𝐶) → ¬ 𝐴 = 𝐶)
9 dfpss2 4059 . 2 (𝐴𝐶 ↔ (𝐴𝐶 ∧ ¬ 𝐴 = 𝐶))
103, 8, 9sylanbrc 583 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wss 3933  wpss 3934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-ne 3014  df-in 3940  df-ss 3949  df-pss 3951
This theorem is referenced by:  sspsstr  4079  psssstr  4080  psstrd  4081  porpss  7442  inf3lem5  9083  ltsopr  10442
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