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Theorem tltnle 28993
 Description: In a Toset, less-than is equivalent to not inverse less-than-or-equal see pltnle 16789. (Contributed by Thierry Arnoux, 11-Feb-2018.)
Hypotheses
Ref Expression
tleile.b 𝐵 = (Base‘𝐾)
tleile.l = (le‘𝐾)
tltnle.s < = (lt‘𝐾)
Assertion
Ref Expression
tltnle ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 𝑋))

Proof of Theorem tltnle
StepHypRef Expression
1 tospos 28989 . . 3 (𝐾 ∈ Toset → 𝐾 ∈ Poset)
2 tleile.b . . . 4 𝐵 = (Base‘𝐾)
3 tleile.l . . . 4 = (le‘𝐾)
4 tltnle.s . . . 4 < = (lt‘𝐾)
52, 3, 4pltval3 16790 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋)))
61, 5syl3an1 1351 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋)))
72, 3tleile 28992 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))
8 ibar 524 . . . 4 ((𝑋 𝑌𝑌 𝑋) → (¬ 𝑌 𝑋 ↔ ((𝑋 𝑌𝑌 𝑋) ∧ ¬ 𝑌 𝑋)))
9 pm5.61 745 . . . 4 (((𝑋 𝑌𝑌 𝑋) ∧ ¬ 𝑌 𝑋) ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋))
108, 9syl6rbb 276 . . 3 ((𝑋 𝑌𝑌 𝑋) → ((𝑋 𝑌 ∧ ¬ 𝑌 𝑋) ↔ ¬ 𝑌 𝑋))
117, 10syl 17 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌 ∧ ¬ 𝑌 𝑋) ↔ ¬ 𝑌 𝑋))
126, 11bitrd 267 1 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 𝑋))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  lecple 15775  Posetcpo 16763  ltcplt 16764  Tosetctos 16856 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-preset 16751  df-poset 16769  df-plt 16781  df-toset 16857 This theorem is referenced by:  tlt2  28995  toslublem  28998  tosglblem  29000  isarchi2  29070  archirng  29073  archiabllem2c  29080  archiabl  29083
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