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Theorem tsrlin 17159
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 17158 . . . 4 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥)))
32simprbi 480 . . 3 (𝑅 ∈ TosetRel → ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥))
4 breq1 4626 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑅𝑦𝐴𝑅𝑦))
5 breq2 4627 . . . . 5 (𝑥 = 𝐴 → (𝑦𝑅𝑥𝑦𝑅𝐴))
64, 5orbi12d 745 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝐴𝑅𝑦𝑦𝑅𝐴)))
7 breq2 4627 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑅𝑦𝐴𝑅𝐵))
8 breq1 4626 . . . . 5 (𝑦 = 𝐵 → (𝑦𝑅𝐴𝐵𝑅𝐴))
97, 8orbi12d 745 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑅𝑦𝑦𝑅𝐴) ↔ (𝐴𝑅𝐵𝐵𝑅𝐴)))
106, 9rspc2v 3311 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥) → (𝐴𝑅𝐵𝐵𝑅𝐴)))
113, 10syl5com 31 . 2 (𝑅 ∈ TosetRel → ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵𝐵𝑅𝐴)))
12113impib 1259 1 ((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908   class class class wbr 4623  dom cdm 5084  PosetRelcps 17138   TosetRel ctsr 17139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-dm 5094  df-tsr 17141
This theorem is referenced by:  tsrlemax  17160  ordtrest2lem  20947  ordthauslem  21127  ordthaus  21128
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