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Theorem sotri3 5064
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 ((𝐶𝑆𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶)

Proof of Theorem sotri3
StepHypRef Expression
1 simp3 1001 . 2 ((𝐶𝑆𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → ¬ 𝐶𝑅𝐵)
2 soi.2 . . . . . 6 𝑅 ⊆ (𝑆 × 𝑆)
32brel 4711 . . . . 5 (𝐴𝑅𝐵 → (𝐴𝑆𝐵𝑆))
433ad2ant2 1021 . . . 4 ((𝐶𝑆𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → (𝐴𝑆𝐵𝑆))
5 simp1 999 . . . 4 ((𝐶𝑆𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐶𝑆)
6 df-3an 982 . . . 4 ((𝐴𝑆𝐵𝑆𝐶𝑆) ↔ ((𝐴𝑆𝐵𝑆) ∧ 𝐶𝑆))
74, 5, 6sylanbrc 417 . . 3 ((𝐶𝑆𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → (𝐴𝑆𝐵𝑆𝐶𝑆))
8 simp2 1000 . . 3 ((𝐶𝑆𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐵)
9 soi.1 . . . 4 𝑅 Or 𝑆
10 sowlin 4351 . . . 4 ((𝑅 Or 𝑆 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → (𝐴𝑅𝐵 → (𝐴𝑅𝐶𝐶𝑅𝐵)))
119, 10mpan 424 . . 3 ((𝐴𝑆𝐵𝑆𝐶𝑆) → (𝐴𝑅𝐵 → (𝐴𝑅𝐶𝐶𝑅𝐵)))
127, 8, 11sylc 62 . 2 ((𝐶𝑆𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → (𝐴𝑅𝐶𝐶𝑅𝐵))
131, 12ecased 1360 1 ((𝐶𝑆𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 709  w3a 980  wcel 2164  wss 3153   class class class wbr 4029   Or wor 4326   × cxp 4657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-iso 4328  df-xp 4665
This theorem is referenced by: (None)
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