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Theorem sotri3 6149
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 5749 . . . 4 (𝐴𝑅𝐵 → (𝐴𝑆𝐵𝑆))
32simprd 495 . . 3 (𝐴𝑅𝐵𝐵𝑆)
4 soi.1 . . . . . . 7 𝑅 Or 𝑆
5 sotric 5621 . . . . . . 7 ((𝑅 Or 𝑆 ∧ (𝐶𝑆𝐵𝑆)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
64, 5mpan 690 . . . . . 6 ((𝐶𝑆𝐵𝑆) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
76con2bid 354 . . . . 5 ((𝐶𝑆𝐵𝑆) → ((𝐶 = 𝐵𝐵𝑅𝐶) ↔ ¬ 𝐶𝑅𝐵))
8 breq2 5146 . . . . . . 7 (𝐶 = 𝐵 → (𝐴𝑅𝐶𝐴𝑅𝐵))
98biimprd 248 . . . . . 6 (𝐶 = 𝐵 → (𝐴𝑅𝐵𝐴𝑅𝐶))
104, 1sotri 6146 . . . . . . 7 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
1110expcom 413 . . . . . 6 (𝐵𝑅𝐶 → (𝐴𝑅𝐵𝐴𝑅𝐶))
129, 11jaoi 857 . . . . 5 ((𝐶 = 𝐵𝐵𝑅𝐶) → (𝐴𝑅𝐵𝐴𝑅𝐶))
137, 12biimtrrdi 254 . . . 4 ((𝐶𝑆𝐵𝑆) → (¬ 𝐶𝑅𝐵 → (𝐴𝑅𝐵𝐴𝑅𝐶)))
1413com3r 87 . . 3 (𝐴𝑅𝐵 → ((𝐶𝑆𝐵𝑆) → (¬ 𝐶𝑅𝐵𝐴𝑅𝐶)))
153, 14mpan2d 694 . 2 (𝐴𝑅𝐵 → (𝐶𝑆 → (¬ 𝐶𝑅𝐵𝐴𝑅𝐶)))
16153imp21 1113 1 ((𝐶𝑆𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1539  wcel 2107  wss 3950   class class class wbr 5142   Or wor 5590   × cxp 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-po 5591  df-so 5592  df-xp 5690
This theorem is referenced by:  archnq  11021
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