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Theorem tltnle 18492
Description: In a Toset, "less than" is equivalent to the negation of the converse of "less than or equal to", see pltnle 18408. (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 18490 . . 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 18409 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋)))
61, 5syl3an1 1163 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋)))
72, 3tleile 18491 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))
8 ibar 528 . . . 4 ((𝑋 𝑌𝑌 𝑋) → (¬ 𝑌 𝑋 ↔ ((𝑋 𝑌𝑌 𝑋) ∧ ¬ 𝑌 𝑋)))
9 pm5.61 1001 . . . 4 (((𝑋 𝑌𝑌 𝑋) ∧ ¬ 𝑌 𝑋) ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋))
108, 9bitr2di 288 . . 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 279 1 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 846  w3a 1087   = wceq 1537  wcel 2108   class class class wbr 5166  cfv 6573  Basecbs 17258  lecple 17318  Posetcpo 18377  ltcplt 18378  Tosetctos 18486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-proset 18365  df-poset 18383  df-plt 18400  df-toset 18487
This theorem is referenced by:  tlt2  32942  toslublem  32945  tosglblem  32947  isarchi2  33165  archirng  33168  archiabllem2c  33175  archiabl  33178
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