Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sotri Structured version   Visualization version   GIF version

Theorem sotri 5960
 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 5587 . . . 4 (𝐴𝑅𝐵 → (𝐴𝑆𝐵𝑆))
32simpld 499 . . 3 (𝐴𝑅𝐵𝐴𝑆)
41brel 5587 . . 3 (𝐵𝑅𝐶 → (𝐵𝑆𝐶𝑆))
53, 4anim12i 616 . 2 ((𝐴𝑅𝐵𝐵𝑅𝐶) → (𝐴𝑆 ∧ (𝐵𝑆𝐶𝑆)))
6 soi.1 . . . 4 𝑅 Or 𝑆
7 sotr 5467 . . . 4 ((𝑅 Or 𝑆 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))
86, 7mpan 690 . . 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 400   ∧ w3a 1085   ∈ wcel 2112   ⊆ wss 3859   class class class wbr 5033   Or wor 5443   × cxp 5523 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-po 5444  df-so 5445  df-xp 5531 This theorem is referenced by:  son2lpi  5961  sotri2  5962  sotri3  5963  ltsonq  10430  ltbtwnnq  10439  nqpr  10475  prlem934  10494  ltexprlem4  10500  reclem2pr  10509  reclem4pr  10511  ltsosr  10555  addgt0sr  10565  supsrlem  10572  axpre-lttrn  10627
 Copyright terms: Public domain W3C validator