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Theorem sotri2 4773
Description: A transitivity relation. (Read ¬ B < A 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
sotri2 ((𝐴𝑆 ∧ ¬ 𝐵𝑅𝐴𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Proof of Theorem sotri2
StepHypRef Expression
1 simp2 940 . 2 ((𝐴𝑆 ∧ ¬ 𝐵𝑅𝐴𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐴)
2 soi.2 . . . . . . 7 𝑅 ⊆ (𝑆 × 𝑆)
32brel 4439 . . . . . 6 (𝐵𝑅𝐶 → (𝐵𝑆𝐶𝑆))
433ad2ant3 962 . . . . 5 ((𝐴𝑆 ∧ ¬ 𝐵𝑅𝐴𝐵𝑅𝐶) → (𝐵𝑆𝐶𝑆))
5 simp1 939 . . . . 5 ((𝐴𝑆 ∧ ¬ 𝐵𝑅𝐴𝐵𝑅𝐶) → 𝐴𝑆)
6 df-3an 922 . . . . 5 ((𝐵𝑆𝐶𝑆𝐴𝑆) ↔ ((𝐵𝑆𝐶𝑆) ∧ 𝐴𝑆))
74, 5, 6sylanbrc 408 . . . 4 ((𝐴𝑆 ∧ ¬ 𝐵𝑅𝐴𝐵𝑅𝐶) → (𝐵𝑆𝐶𝑆𝐴𝑆))
8 simp3 941 . . . 4 ((𝐴𝑆 ∧ ¬ 𝐵𝑅𝐴𝐵𝑅𝐶) → 𝐵𝑅𝐶)
9 soi.1 . . . . 5 𝑅 Or 𝑆
10 sowlin 4104 . . . . 5 ((𝑅 Or 𝑆 ∧ (𝐵𝑆𝐶𝑆𝐴𝑆)) → (𝐵𝑅𝐶 → (𝐵𝑅𝐴𝐴𝑅𝐶)))
119, 10mpan 415 . . . 4 ((𝐵𝑆𝐶𝑆𝐴𝑆) → (𝐵𝑅𝐶 → (𝐵𝑅𝐴𝐴𝑅𝐶)))
127, 8, 11sylc 61 . . 3 ((𝐴𝑆 ∧ ¬ 𝐵𝑅𝐴𝐵𝑅𝐶) → (𝐵𝑅𝐴𝐴𝑅𝐶))
1312ord 676 . 2 ((𝐴𝑆 ∧ ¬ 𝐵𝑅𝐴𝐵𝑅𝐶) → (¬ 𝐵𝑅𝐴𝐴𝑅𝐶))
141, 13mpd 13 1 ((𝐴𝑆 ∧ ¬ 𝐵𝑅𝐴𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 662  w3a 920  wcel 1434  wss 2983   class class class wbr 3806   Or wor 4079   × cxp 4390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2613  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-br 3807  df-opab 3861  df-iso 4081  df-xp 4398
This theorem is referenced by: (None)
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