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Theorem tsrlin 17880
 Description: A toset is a linear order. (Contributed by Mario Carneiro, 9-Sep-2015.)
Hypothesis
Ref Expression
istsr.1 𝑋 = dom 𝑅
Assertion
Ref Expression
tsrlin ((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵𝐵𝑅𝐴))

Proof of Theorem tsrlin
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istsr.1 . . . . 5 𝑋 = dom 𝑅
21istsr2 17879 . . . 4 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥)))
32simprbi 501 . . 3 (𝑅 ∈ TosetRel → ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥))
4 breq1 5028 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑅𝑦𝐴𝑅𝑦))
5 breq2 5029 . . . . 5 (𝑥 = 𝐴 → (𝑦𝑅𝑥𝑦𝑅𝐴))
64, 5orbi12d 917 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝐴𝑅𝑦𝑦𝑅𝐴)))
7 breq2 5029 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑅𝑦𝐴𝑅𝐵))
8 breq1 5028 . . . . 5 (𝑦 = 𝐵 → (𝑦𝑅𝐴𝐵𝑅𝐴))
97, 8orbi12d 917 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑅𝑦𝑦𝑅𝐴) ↔ (𝐴𝑅𝐵𝐵𝑅𝐴)))
106, 9rspc2v 3549 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥) → (𝐴𝑅𝐵𝐵𝑅𝐴)))
113, 10syl5com 31 . 2 (𝑅 ∈ TosetRel → ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵𝐵𝑅𝐴)))
12113impib 1114 1 ((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵𝐵𝑅𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 845   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3068   class class class wbr 5025  dom cdm 5517  PosetRelcps 17859   TosetRel ctsr 17860 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 5162  ax-nul 5169  ax-pr 5291 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ral 3073  df-rex 3074  df-rab 3077  df-v 3409  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-nul 4222  df-if 4414  df-sn 4516  df-pr 4518  df-op 4522  df-br 5026  df-opab 5088  df-xp 5523  df-rel 5524  df-cnv 5525  df-dm 5527  df-tsr 17862 This theorem is referenced by:  tsrlemax  17881  ordtrest2lem  21888  ordthauslem  22068  ordthaus  22069
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