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Theorem psstr 3373
 Description: Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
Assertion
Ref Expression
psstr ((AB BC) → AC)

Proof of Theorem psstr
StepHypRef Expression
1 pssss 3364 . . 3 (ABA B)
2 pssss 3364 . . 3 (BCB C)
31, 2sylan9ss 3285 . 2 ((AB BC) → A C)
4 pssn2lp 3370 . . . 4 ¬ (CB BC)
5 psseq1 3356 . . . . 5 (A = C → (ABCB))
65anbi1d 685 . . . 4 (A = C → ((AB BC) ↔ (CB BC)))
74, 6mtbiri 294 . . 3 (A = C → ¬ (AB BC))
87con2i 112 . 2 ((AB BC) → ¬ A = C)
9 dfpss2 3354 . 2 (AC ↔ (A C ¬ A = C))
103, 8, 9sylanbrc 645 1 ((AB BC) → AC)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ⊆ wss 3257   ⊊ wpss 3258 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pss 3261 This theorem is referenced by:  sspsstr  3374  psssstr  3375  psstrd  3376
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