Theorem tltnle 18357

Description: In a Toset, "less than" is equivalent to the negation of the converse of "less than or equal to", see pltnle 18273. (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 18355	. . 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 18274	. . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋)))
6	1, 5	syl3an1 1164	. 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋)))
7	2, 3	tleile 18356	. . 3 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋))
8		ibar 528	. . . 4 ⊢ ((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) → (¬ 𝑌 ≤ 𝑋 ↔ ((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) ∧ ¬ 𝑌 ≤ 𝑋)))
9		pm5.61 1003	. . . 4 ⊢ (((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) ∧ ¬ 𝑌 ≤ 𝑋) ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋))
10	8, 9	bitr2di 288	. . 3 ⊢ ((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) → ((𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋) ↔ ¬ 𝑌 ≤ 𝑋))
11	7, 10	syl 17	. 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋) ↔ ¬ 𝑌 ≤ 𝑋))
12	6, 11	bitrd 279	1 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 ≤ 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   class class class wbr 5100  ‘cfv 6502  Basecbs 17150  lecple 17198  Posetcpo 18244  ltcplt 18245  Tosetctos 18351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6458  df-fun 6504  df-fv 6510  df-proset 18231  df-poset 18250  df-plt 18265  df-toset 18352

This theorem is referenced by:  tlt2  33068  toslublem  33071  tosglblem  33073  isarchi2  33285  archirng  33288  archiabllem2c  33295  archiabl  33298