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Theorem sotri3 5524
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 5166 . . . 4 (𝐴𝑅𝐵 → (𝐴𝑆𝐵𝑆))
32simprd 479 . . 3 (𝐴𝑅𝐵𝐵𝑆)
4 soi.1 . . . . . . 7 𝑅 Or 𝑆
5 sotric 5059 . . . . . . 7 ((𝑅 Or 𝑆 ∧ (𝐶𝑆𝐵𝑆)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
64, 5mpan 706 . . . . . 6 ((𝐶𝑆𝐵𝑆) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
76con2bid 344 . . . . 5 ((𝐶𝑆𝐵𝑆) → ((𝐶 = 𝐵𝐵𝑅𝐶) ↔ ¬ 𝐶𝑅𝐵))
8 breq2 4655 . . . . . . 7 (𝐶 = 𝐵 → (𝐴𝑅𝐶𝐴𝑅𝐵))
98biimprd 238 . . . . . 6 (𝐶 = 𝐵 → (𝐴𝑅𝐵𝐴𝑅𝐶))
104, 1sotri 5521 . . . . . . 7 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
1110expcom 451 . . . . . 6 (𝐵𝑅𝐶 → (𝐴𝑅𝐵𝐴𝑅𝐶))
129, 11jaoi 394 . . . . 5 ((𝐶 = 𝐵𝐵𝑅𝐶) → (𝐴𝑅𝐵𝐴𝑅𝐶))
137, 12syl6bir 244 . . . 4 ((𝐶𝑆𝐵𝑆) → (¬ 𝐶𝑅𝐵 → (𝐴𝑅𝐵𝐴𝑅𝐶)))
1413com3r 87 . . 3 (𝐴𝑅𝐵 → ((𝐶𝑆𝐵𝑆) → (¬ 𝐶𝑅𝐵𝐴𝑅𝐶)))
153, 14mpan2d 710 . 2 (𝐴𝑅𝐵 → (𝐶𝑆 → (¬ 𝐶𝑅𝐵𝐴𝑅𝐶)))
16153imp21 1276 1 ((𝐶𝑆𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1037   = wceq 1482  wcel 1989  wss 3572   class class class wbr 4651   Or wor 5032   × cxp 5110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-po 5033  df-so 5034  df-xp 5118
This theorem is referenced by:  archnq  9799
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