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Theorem sotri3 4666
Description: A transitivity relation. (Read A < B and ¬ C < B 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 ((𝐶 𝑆 A𝑅B ¬ 𝐶𝑅B) → A𝑅𝐶)

Proof of Theorem sotri3
StepHypRef Expression
1 simp3 905 . 2 ((𝐶 𝑆 A𝑅B ¬ 𝐶𝑅B) → ¬ 𝐶𝑅B)
2 soi.2 . . . . . 6 𝑅 ⊆ (𝑆 × 𝑆)
32brel 4335 . . . . 5 (A𝑅B → (A 𝑆 B 𝑆))
433ad2ant2 925 . . . 4 ((𝐶 𝑆 A𝑅B ¬ 𝐶𝑅B) → (A 𝑆 B 𝑆))
5 simp1 903 . . . 4 ((𝐶 𝑆 A𝑅B ¬ 𝐶𝑅B) → 𝐶 𝑆)
6 df-3an 886 . . . 4 ((A 𝑆 B 𝑆 𝐶 𝑆) ↔ ((A 𝑆 B 𝑆) 𝐶 𝑆))
74, 5, 6sylanbrc 394 . . 3 ((𝐶 𝑆 A𝑅B ¬ 𝐶𝑅B) → (A 𝑆 B 𝑆 𝐶 𝑆))
8 simp2 904 . . 3 ((𝐶 𝑆 A𝑅B ¬ 𝐶𝑅B) → A𝑅B)
9 soi.1 . . . 4 𝑅 Or 𝑆
10 sowlin 4048 . . . 4 ((𝑅 Or 𝑆 (A 𝑆 B 𝑆 𝐶 𝑆)) → (A𝑅B → (A𝑅𝐶 𝐶𝑅B)))
119, 10mpan 400 . . 3 ((A 𝑆 B 𝑆 𝐶 𝑆) → (A𝑅B → (A𝑅𝐶 𝐶𝑅B)))
127, 8, 11sylc 56 . 2 ((𝐶 𝑆 A𝑅B ¬ 𝐶𝑅B) → (A𝑅𝐶 𝐶𝑅B))
131, 12ecased 1238 1 ((𝐶 𝑆 A𝑅B ¬ 𝐶𝑅B) → A𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wo 628   w3a 884   wcel 1390  wss 2911   class class class wbr 3755   Or wor 4023   × cxp 4286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-iso 4025  df-xp 4294
This theorem is referenced by: (None)
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