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Theorem sotri3 5742
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 5369 . . . 4 (𝐴𝑅𝐵 → (𝐴𝑆𝐵𝑆))
32simprd 490 . . 3 (𝐴𝑅𝐵𝐵𝑆)
4 soi.1 . . . . . . 7 𝑅 Or 𝑆
5 sotric 5257 . . . . . . 7 ((𝑅 Or 𝑆 ∧ (𝐶𝑆𝐵𝑆)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
64, 5mpan 682 . . . . . 6 ((𝐶𝑆𝐵𝑆) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
76con2bid 346 . . . . 5 ((𝐶𝑆𝐵𝑆) → ((𝐶 = 𝐵𝐵𝑅𝐶) ↔ ¬ 𝐶𝑅𝐵))
8 breq2 4845 . . . . . . 7 (𝐶 = 𝐵 → (𝐴𝑅𝐶𝐴𝑅𝐵))
98biimprd 240 . . . . . 6 (𝐶 = 𝐵 → (𝐴𝑅𝐵𝐴𝑅𝐶))
104, 1sotri 5739 . . . . . . 7 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
1110expcom 403 . . . . . 6 (𝐵𝑅𝐶 → (𝐴𝑅𝐵𝐴𝑅𝐶))
129, 11jaoi 884 . . . . 5 ((𝐶 = 𝐵𝐵𝑅𝐶) → (𝐴𝑅𝐵𝐴𝑅𝐶))
137, 12syl6bir 246 . . . 4 ((𝐶𝑆𝐵𝑆) → (¬ 𝐶𝑅𝐵 → (𝐴𝑅𝐵𝐴𝑅𝐶)))
1413com3r 87 . . 3 (𝐴𝑅𝐵 → ((𝐶𝑆𝐵𝑆) → (¬ 𝐶𝑅𝐵𝐴𝑅𝐶)))
153, 14mpan2d 686 . 2 (𝐴𝑅𝐵 → (𝐶𝑆 → (¬ 𝐶𝑅𝐵𝐴𝑅𝐶)))
16153imp21 1142 1 ((𝐶𝑆𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  wo 874  w3a 1108   = wceq 1653  wcel 2157  wss 3767   class class class wbr 4841   Or wor 5230   × cxp 5308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-po 5231  df-so 5232  df-xp 5316
This theorem is referenced by:  archnq  10088
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