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Theorem sotri2 5427
Description: A transitivity relation. (Read 𝐴𝐵 and 𝐵 < 𝐶 implies 𝐴 < 𝐶.) (Contributed by Mario Carneiro, 10-May-2013.)
Hypotheses
Ref Expression
soi.1 𝑅 Or 𝑆
soi.2 𝑅 ⊆ (𝑆 × 𝑆)
Assertion
Ref Expression
sotri2 ((𝐴𝑆 ∧ ¬ 𝐵𝑅𝐴𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Proof of Theorem sotri2
StepHypRef Expression
1 soi.2 . . . . . 6 𝑅 ⊆ (𝑆 × 𝑆)
21brel 5076 . . . . 5 (𝐵𝑅𝐶 → (𝐵𝑆𝐶𝑆))
32simpld 473 . . . 4 (𝐵𝑅𝐶𝐵𝑆)
4 soi.1 . . . . . . . 8 𝑅 Or 𝑆
5 sotric 4971 . . . . . . . 8 ((𝑅 Or 𝑆 ∧ (𝐵𝑆𝐴𝑆)) → (𝐵𝑅𝐴 ↔ ¬ (𝐵 = 𝐴𝐴𝑅𝐵)))
64, 5mpan 701 . . . . . . 7 ((𝐵𝑆𝐴𝑆) → (𝐵𝑅𝐴 ↔ ¬ (𝐵 = 𝐴𝐴𝑅𝐵)))
76con2bid 342 . . . . . 6 ((𝐵𝑆𝐴𝑆) → ((𝐵 = 𝐴𝐴𝑅𝐵) ↔ ¬ 𝐵𝑅𝐴))
8 breq1 4576 . . . . . . . 8 (𝐵 = 𝐴 → (𝐵𝑅𝐶𝐴𝑅𝐶))
98biimpd 217 . . . . . . 7 (𝐵 = 𝐴 → (𝐵𝑅𝐶𝐴𝑅𝐶))
104, 1sotri 5425 . . . . . . . 8 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
1110ex 448 . . . . . . 7 (𝐴𝑅𝐵 → (𝐵𝑅𝐶𝐴𝑅𝐶))
129, 11jaoi 392 . . . . . 6 ((𝐵 = 𝐴𝐴𝑅𝐵) → (𝐵𝑅𝐶𝐴𝑅𝐶))
137, 12syl6bir 242 . . . . 5 ((𝐵𝑆𝐴𝑆) → (¬ 𝐵𝑅𝐴 → (𝐵𝑅𝐶𝐴𝑅𝐶)))
1413com3r 84 . . . 4 (𝐵𝑅𝐶 → ((𝐵𝑆𝐴𝑆) → (¬ 𝐵𝑅𝐴𝐴𝑅𝐶)))
153, 14mpand 706 . . 3 (𝐵𝑅𝐶 → (𝐴𝑆 → (¬ 𝐵𝑅𝐴𝐴𝑅𝐶)))
1615com3l 86 . 2 (𝐴𝑆 → (¬ 𝐵𝑅𝐴 → (𝐵𝑅𝐶𝐴𝑅𝐶)))
17163imp 1248 1 ((𝐴𝑆 ∧ ¬ 𝐵𝑅𝐴𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1975  wss 3535   class class class wbr 4573   Or wor 4944   × cxp 5022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-br 4574  df-opab 4634  df-po 4945  df-so 4946  df-xp 5030
This theorem is referenced by:  supsrlem  9784
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