Theorem tltnle 18482

Description: In a Toset, "less than" is equivalent to the negation of the converse of "less than or equal to", see pltnle 18398. (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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 ≤ 𝑋))

Proof of Theorem tltnle

Step	Hyp	Ref	Expression
1		tospos 18480	. . 3 ⊢ (𝐾 ∈ Toset → 𝐾 ∈ Poset)
2		tleile.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
3		tleile.l	. . . 4 ⊢ ≤ = (le‘𝐾)
4		tltnle.s	. . . 4 ⊢ < = (lt‘𝐾)
5	2, 3, 4	pltval3 18399	. . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋)))
6	1, 5	syl3an1 1180	. 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋)))
7	2, 3	tleile 18481	. . 3 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋))
8		ibar 537	. . . 4 ⊢ ((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) → (¬ 𝑌 ≤ 𝑋 ↔ ((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) ∧ ¬ 𝑌 ≤ 𝑋)))
9		pm5.61 1015	. . . 4 ⊢ (((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) ∧ ¬ 𝑌 ≤ 𝑋) ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋))
10	8, 9	bitr2di 291	. . 3 ⊢ ((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) → ((𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋) ↔ ¬ 𝑌 ≤ 𝑋))
11	7, 10	syl 18	. 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋) ↔ ¬ 𝑌 ≤ 𝑋))
12	6, 11	bitrd 282	1 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 ≤ 𝑋))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   class class class wbr 5108  ‘cfv 6536  Basecbs 17275  lecple 17323  Posetcpo 18369  ltcplt 18370  Tosetctos 18476

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-proset 18356  df-poset 18375  df-plt 18390  df-toset 18477

This theorem is used by:  tlt2  33298  toslublem  33301  tosglblem  33303  isarchi2  33514  archirng  33517  archiabllem2c  33524  archiabl  33527