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Theorem tsrlin 18434
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 18433 . . . 4 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥)))
32simprbi 498 . . 3 (𝑅 ∈ TosetRel → ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥))
4 breq1 5107 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑅𝑦𝐴𝑅𝑦))
5 breq2 5108 . . . . 5 (𝑥 = 𝐴 → (𝑦𝑅𝑥𝑦𝑅𝐴))
64, 5orbi12d 918 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝐴𝑅𝑦𝑦𝑅𝐴)))
7 breq2 5108 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑅𝑦𝐴𝑅𝐵))
8 breq1 5107 . . . . 5 (𝑦 = 𝐵 → (𝑦𝑅𝐴𝐵𝑅𝐴))
97, 8orbi12d 918 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑅𝑦𝑦𝑅𝐴) ↔ (𝐴𝑅𝐵𝐵𝑅𝐴)))
106, 9rspc2v 3589 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥) → (𝐴𝑅𝐵𝐵𝑅𝐴)))
113, 10syl5com 31 . 2 (𝑅 ∈ TosetRel → ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵𝐵𝑅𝐴)))
12113impib 1117 1 ((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wral 3063   class class class wbr 5104  dom cdm 5632  PosetRelcps 18413   TosetRel ctsr 18414
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 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
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 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-tsr 18416
This theorem is referenced by:  tsrlemax  18435  ordtrest2lem  22506  ordthauslem  22686  ordthaus  22687
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