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Theorem sotri3 5961
 Description: A transitivity relation. (Read 𝐴 < 𝐵 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
sotri3 ((𝐶𝑆𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶)

Proof of Theorem sotri3
StepHypRef Expression
1 soi.2 . . . . 5 𝑅 ⊆ (𝑆 × 𝑆)
21brel 5585 . . . 4 (𝐴𝑅𝐵 → (𝐴𝑆𝐵𝑆))
32simprd 499 . . 3 (𝐴𝑅𝐵𝐵𝑆)
4 soi.1 . . . . . . 7 𝑅 Or 𝑆
5 sotric 5469 . . . . . . 7 ((𝑅 Or 𝑆 ∧ (𝐶𝑆𝐵𝑆)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
64, 5mpan 689 . . . . . 6 ((𝐶𝑆𝐵𝑆) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
76con2bid 358 . . . . 5 ((𝐶𝑆𝐵𝑆) → ((𝐶 = 𝐵𝐵𝑅𝐶) ↔ ¬ 𝐶𝑅𝐵))
8 breq2 5038 . . . . . . 7 (𝐶 = 𝐵 → (𝐴𝑅𝐶𝐴𝑅𝐵))
98biimprd 251 . . . . . 6 (𝐶 = 𝐵 → (𝐴𝑅𝐵𝐴𝑅𝐶))
104, 1sotri 5958 . . . . . . 7 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
1110expcom 417 . . . . . 6 (𝐵𝑅𝐶 → (𝐴𝑅𝐵𝐴𝑅𝐶))
129, 11jaoi 854 . . . . 5 ((𝐶 = 𝐵𝐵𝑅𝐶) → (𝐴𝑅𝐵𝐴𝑅𝐶))
137, 12syl6bir 257 . . . 4 ((𝐶𝑆𝐵𝑆) → (¬ 𝐶𝑅𝐵 → (𝐴𝑅𝐵𝐴𝑅𝐶)))
1413com3r 87 . . 3 (𝐴𝑅𝐵 → ((𝐶𝑆𝐵𝑆) → (¬ 𝐶𝑅𝐵𝐴𝑅𝐶)))
153, 14mpan2d 693 . 2 (𝐴𝑅𝐵 → (𝐶𝑆 → (¬ 𝐶𝑅𝐵𝐴𝑅𝐶)))
16153imp21 1111 1 ((𝐶𝑆𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ⊆ wss 3883   class class class wbr 5034   Or wor 5441   × cxp 5521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-po 5442  df-so 5443  df-xp 5529 This theorem is referenced by:  archnq  10409
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