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Theorem sotri2 4649
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 ((A 𝑆 ¬ B𝑅A B𝑅𝐶) → A𝑅𝐶)

Proof of Theorem sotri2
StepHypRef Expression
1 simp2 893 . 2 ((A 𝑆 ¬ B𝑅A B𝑅𝐶) → ¬ B𝑅A)
2 soi.2 . . . . . . 7 𝑅 ⊆ (𝑆 × 𝑆)
32brel 4319 . . . . . 6 (B𝑅𝐶 → (B 𝑆 𝐶 𝑆))
433ad2ant3 915 . . . . 5 ((A 𝑆 ¬ B𝑅A B𝑅𝐶) → (B 𝑆 𝐶 𝑆))
5 simp1 892 . . . . 5 ((A 𝑆 ¬ B𝑅A B𝑅𝐶) → A 𝑆)
6 df-3an 875 . . . . 5 ((B 𝑆 𝐶 𝑆 A 𝑆) ↔ ((B 𝑆 𝐶 𝑆) A 𝑆))
74, 5, 6sylanbrc 396 . . . 4 ((A 𝑆 ¬ B𝑅A B𝑅𝐶) → (B 𝑆 𝐶 𝑆 A 𝑆))
8 simp3 894 . . . 4 ((A 𝑆 ¬ B𝑅A B𝑅𝐶) → B𝑅𝐶)
9 soi.1 . . . . 5 𝑅 Or 𝑆
10 sowlin 4031 . . . . 5 ((𝑅 Or 𝑆 (B 𝑆 𝐶 𝑆 A 𝑆)) → (B𝑅𝐶 → (B𝑅A A𝑅𝐶)))
119, 10mpan 402 . . . 4 ((B 𝑆 𝐶 𝑆 A 𝑆) → (B𝑅𝐶 → (B𝑅A A𝑅𝐶)))
127, 8, 11sylc 56 . . 3 ((A 𝑆 ¬ B𝑅A B𝑅𝐶) → (B𝑅A A𝑅𝐶))
1312ord 630 . 2 ((A 𝑆 ¬ B𝑅A B𝑅𝐶) → (¬ B𝑅AA𝑅𝐶))
141, 13mpd 13 1 ((A 𝑆 ¬ B𝑅A B𝑅𝐶) → A𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wo 616   w3a 873   wcel 1374  wss 2894   class class class wbr 3738   Or wor 4006   × cxp 4270
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 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-iso 4008  df-xp 4278
This theorem is referenced by: (None)
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