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Theorem sotri3 5428
Description: A transitivity relation. (Read 𝐴 < 𝐵 and 𝐵𝐶 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
sotri3 ((𝐶𝑆𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶)

Proof of Theorem sotri3
StepHypRef Expression
1 soi.2 . . . . 5 𝑅 ⊆ (𝑆 × 𝑆)
21brel 5076 . . . 4 (𝐴𝑅𝐵 → (𝐴𝑆𝐵𝑆))
32simprd 477 . . 3 (𝐴𝑅𝐵𝐵𝑆)
4 soi.1 . . . . . . 7 𝑅 Or 𝑆
5 sotric 4971 . . . . . . 7 ((𝑅 Or 𝑆 ∧ (𝐶𝑆𝐵𝑆)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
64, 5mpan 701 . . . . . 6 ((𝐶𝑆𝐵𝑆) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
76con2bid 342 . . . . 5 ((𝐶𝑆𝐵𝑆) → ((𝐶 = 𝐵𝐵𝑅𝐶) ↔ ¬ 𝐶𝑅𝐵))
8 breq2 4577 . . . . . . 7 (𝐶 = 𝐵 → (𝐴𝑅𝐶𝐴𝑅𝐵))
98biimprd 236 . . . . . 6 (𝐶 = 𝐵 → (𝐴𝑅𝐵𝐴𝑅𝐶))
104, 1sotri 5425 . . . . . . 7 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
1110expcom 449 . . . . . 6 (𝐵𝑅𝐶 → (𝐴𝑅𝐵𝐴𝑅𝐶))
129, 11jaoi 392 . . . . 5 ((𝐶 = 𝐵𝐵𝑅𝐶) → (𝐴𝑅𝐵𝐴𝑅𝐶))
137, 12syl6bir 242 . . . 4 ((𝐶𝑆𝐵𝑆) → (¬ 𝐶𝑅𝐵 → (𝐴𝑅𝐵𝐴𝑅𝐶)))
1413com3r 84 . . 3 (𝐴𝑅𝐵 → ((𝐶𝑆𝐵𝑆) → (¬ 𝐶𝑅𝐵𝐴𝑅𝐶)))
153, 14mpan2d 705 . 2 (𝐴𝑅𝐵 → (𝐶𝑆 → (¬ 𝐶𝑅𝐵𝐴𝑅𝐶)))
16153imp21 1268 1 ((𝐶𝑆𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 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:  archnq  9654
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