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Theorem sotri 6085
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 5701 . . . 4 (𝐴𝑅𝐵 → (𝐴𝑆𝐵𝑆))
32simpld 496 . . 3 (𝐴𝑅𝐵𝐴𝑆)
41brel 5701 . . 3 (𝐵𝑅𝐶 → (𝐵𝑆𝐶𝑆))
53, 4anim12i 614 . 2 ((𝐴𝑅𝐵𝐵𝑅𝐶) → (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
6 soi.1 . . . 4 𝑅 Or 𝑆
7 sotr 5573 . . . 4 ((𝑅 Or 𝑆 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
86, 7mpan 689 . . 3 ((𝐴𝑆𝐵𝑆𝐶𝑆) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
983expb 1121 . 2 ((𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
105, 9mpcom 38 1 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088  wcel 2107  wss 3914   class class class wbr 5109   Or wor 5548   × cxp 5635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-po 5549  df-so 5550  df-xp 5643
This theorem is referenced by:  son2lpi  6086  sotri2  6087  sotri3  6088  ltsonq  10913  ltbtwnnq  10922  nqpr  10958  prlem934  10977  ltexprlem4  10983  reclem2pr  10992  reclem4pr  10994  ltsosr  11038  addgt0sr  11048  supsrlem  11055  axpre-lttrn  11110
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