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Theorem sotri 6127
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 5737 . . . 4 (𝐴𝑅𝐵 → (𝐴𝑆𝐵𝑆))
32simpld 494 . . 3 (𝐴𝑅𝐵𝐴𝑆)
41brel 5737 . . 3 (𝐵𝑅𝐶 → (𝐵𝑆𝐶𝑆))
53, 4anim12i 612 . 2 ((𝐴𝑅𝐵𝐵𝑅𝐶) → (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
6 soi.1 . . . 4 𝑅 Or 𝑆
7 sotr 5608 . . . 4 ((𝑅 Or 𝑆 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
86, 7mpan 689 . . 3 ((𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
983expb 1118 . 2 ((𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
105, 9mpcom 38 1 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085  wcel 2099  wss 3945   class class class wbr 5142   Or wor 5583   × cxp 5670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-po 5584  df-so 5585  df-xp 5678
This theorem is referenced by:  son2lpi  6128  sotri2  6129  sotri3  6130  ltsonq  10986  ltbtwnnq  10995  nqpr  11031  prlem934  11050  ltexprlem4  11056  reclem2pr  11065  reclem4pr  11067  ltsosr  11111  addgt0sr  11121  supsrlem  11128  axpre-lttrn  11183
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