Theorem tltnle 18055

Description: In a Toset, "less than" is equivalent to the negation of the converse of "less than or equal to", see pltnle 17971. (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 18053	. . 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 17972	. . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋)))
6	1, 5	syl3an1 1161	. 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋)))
7	2, 3	tleile 18054	. . 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		pm5.61 997	. . . 4 ⊢ (((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) ∧ ¬ 𝑌 ≤ 𝑋) ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋))
10	8, 9	bitr2di 287	. . 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 278	1 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 ≤ 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   class class class wbr 5070  ‘cfv 6418  Basecbs 16840  lecple 16895  Posetcpo 17940  ltcplt 17941  Tosetctos 18049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-proset 17928  df-poset 17946  df-plt 17963  df-toset 18050

This theorem is referenced by:  tlt2  31149  toslublem  31152  tosglblem  31154  isarchi2  31341  archirng  31344  archiabllem2c  31351  archiabl  31354