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Theorem sotri 6118
Description: A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
Hypotheses
Ref Expression
soi.1 𝑅 Or 𝑆
soi.2 𝑅 ⊆ (𝑆 × 𝑆)
Assertion
Ref Expression
sotri ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Proof of Theorem sotri
StepHypRef Expression
1 soi.2 . . . . 5 𝑅 ⊆ (𝑆 × 𝑆)
21brel 5731 . . . 4 (𝐴𝑅𝐵 → (𝐴𝑆𝐵𝑆))
32simpld 494 . . 3 (𝐴𝑅𝐵𝐴𝑆)
41brel 5731 . . 3 (𝐵𝑅𝐶 → (𝐵𝑆𝐶𝑆))
53, 4anim12i 612 . 2 ((𝐴𝑅𝐵𝐵𝑅𝐶) → (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
6 soi.1 . . . 4 𝑅 Or 𝑆
7 sotr 5602 . . . 4 ((𝑅 Or 𝑆 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
86, 7mpan 687 . . 3 ((𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
983expb 1117 . 2 ((𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
105, 9mpcom 38 1 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084  wcel 2098  wss 3940   class class class wbr 5138   Or wor 5577   × cxp 5664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-po 5578  df-so 5579  df-xp 5672
This theorem is referenced by:  son2lpi  6119  sotri2  6120  sotri3  6121  ltsonq  10960  ltbtwnnq  10969  nqpr  11005  prlem934  11024  ltexprlem4  11030  reclem2pr  11039  reclem4pr  11041  ltsosr  11085  addgt0sr  11095  supsrlem  11102  axpre-lttrn  11157
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