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Theorem tsrlin 17817
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 17816 . . . 4 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥)))
32simprbi 497 . . 3 (𝑅 ∈ TosetRel → ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥))
4 breq1 5060 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑅𝑦𝐴𝑅𝑦))
5 breq2 5061 . . . . 5 (𝑥 = 𝐴 → (𝑦𝑅𝑥𝑦𝑅𝐴))
64, 5orbi12d 912 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝐴𝑅𝑦𝑦𝑅𝐴)))
7 breq2 5061 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑅𝑦𝐴𝑅𝐵))
8 breq1 5060 . . . . 5 (𝑦 = 𝐵 → (𝑦𝑅𝐴𝐵𝑅𝐴))
97, 8orbi12d 912 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑅𝑦𝑦𝑅𝐴) ↔ (𝐴𝑅𝐵𝐵𝑅𝐴)))
106, 9rspc2v 3630 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥) → (𝐴𝑅𝐵𝐵𝑅𝐴)))
113, 10syl5com 31 . 2 (𝑅 ∈ TosetRel → ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵𝐵𝑅𝐴)))
12113impib 1108 1 ((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wral 3135   class class class wbr 5057  dom cdm 5548  PosetRelcps 17796   TosetRel ctsr 17797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-tsr 17799
This theorem is referenced by:  tsrlemax  17818  ordtrest2lem  21739  ordthauslem  21919  ordthaus  21920
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