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Theorem tsrlin 17440
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 17439 . . . 4 (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥)))
32simprbi 483 . . 3 (𝑅 ∈ TosetRel → ∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥))
4 breq1 4807 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑅𝑦𝐴𝑅𝑦))
5 breq2 4808 . . . . 5 (𝑥 = 𝐴 → (𝑦𝑅𝑥𝑦𝑅𝐴))
64, 5orbi12d 748 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ↔ (𝐴𝑅𝑦𝑦𝑅𝐴)))
7 breq2 4808 . . . . 5 (𝑦 = 𝐵 → (𝐴𝑅𝑦𝐴𝑅𝐵))
8 breq1 4807 . . . . 5 (𝑦 = 𝐵 → (𝑦𝑅𝐴𝐵𝑅𝐴))
97, 8orbi12d 748 . . . 4 (𝑦 = 𝐵 → ((𝐴𝑅𝑦𝑦𝑅𝐴) ↔ (𝐴𝑅𝐵𝐵𝑅𝐴)))
106, 9rspc2v 3461 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑅𝑦𝑦𝑅𝑥) → (𝐴𝑅𝐵𝐵𝑅𝐴)))
113, 10syl5com 31 . 2 (𝑅 ∈ TosetRel → ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵𝐵𝑅𝐴)))
12113impib 1109 1 ((𝑅 ∈ TosetRel ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050   class class class wbr 4804  dom cdm 5266  PosetRelcps 17419   TosetRel ctsr 17420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-tsr 17422
This theorem is referenced by:  tsrlemax  17441  ordtrest2lem  21229  ordthauslem  21409  ordthaus  21410
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