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Theorem tltnle 18467
Description: In a Toset, "less than" is equivalent to the negation of the converse of "less than or equal to", see pltnle 18383. (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 18465 . . 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 18384 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋)))
61, 5syl3an1 1164 . 2 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌 ∧ ¬ 𝑌 𝑋)))
72, 3tleile 18466 . . 3 ((𝐾 ∈ Toset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 𝑋))
8 ibar 528 . . . 4 ((𝑋 𝑌𝑌 𝑋) → (¬ 𝑌 𝑋 ↔ ((𝑋 𝑌𝑌 𝑋) ∧ ¬ 𝑌 𝑋)))
9 pm5.61 1003 . . . 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 848  w3a 1087   = wceq 1540  wcel 2108   class class class wbr 5143  cfv 6561  Basecbs 17247  lecple 17304  Posetcpo 18353  ltcplt 18354  Tosetctos 18461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-proset 18340  df-poset 18359  df-plt 18375  df-toset 18462
This theorem is referenced by:  tlt2  32959  toslublem  32962  tosglblem  32964  isarchi2  33192  archirng  33195  archiabllem2c  33202  archiabl  33205
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