Theorem tleile 18322

Description: In a Toset, any two elements are comparable. (Contributed by Thierry Arnoux, 11-Feb-2018.)

Hypotheses

Ref	Expression
tleile.b	⊢ 𝐵 = (Base‘𝐾)
tleile.l	⊢ ≤ = (le‘𝐾)

Assertion

Ref	Expression
tleile	⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋))

Proof of Theorem tleile

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1137	. 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
2		simp3 1138	. 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
3		tleile.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
4		tleile.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
5	3, 4	istos 18319	. . . 4 ⊢ (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
6	5	simprbi 496	. . 3 ⊢ (𝐾 ∈ Toset → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
7	6	3ad2ant1 1133	. 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
8		breq1 5094	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
9		breq2 5095	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑋))
10	8, 9	orbi12d 918	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ (𝑋 ≤ 𝑦 ∨ 𝑦 ≤ 𝑋)))
11		breq2 5095	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
12		breq1 5094	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 ≤ 𝑋 ↔ 𝑌 ≤ 𝑋))
13	11, 12	orbi12d 918	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 ∨ 𝑦 ≤ 𝑋) ↔ (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋)))
14	10, 13	rspc2va 3589	. 2 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋))
15	1, 2, 7, 14	syl21anc 837	1 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋))

Syntax hints:   → wi 4   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047   class class class wbr 5091  ‘cfv 6481  Basecbs 17117  lecple 17165  Posetcpo 18210  Tosetctos 18317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5244

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-iota 6437  df-fv 6489  df-toset 18318

This theorem is referenced by:  tltnle  18323  odutos  32944  trleile  32947  toslat  49012