Theorem tleile 18479

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 1136	. 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
2		simp3 1137	. 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 18476	. . . 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 1132	. 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
8		breq1 5151	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
9		breq2 5152	. . . 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 5152	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
12		breq1 5151	. . . 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 3634	. 2 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋))
15	1, 2, 7, 14	syl21anc 838	1 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋))

Syntax hints:   → wi 4   ∨ wo 847   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059   class class class wbr 5148  ‘cfv 6563  Basecbs 17245  lecple 17305  Posetcpo 18365  Tosetctos 18474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-toset 18475

This theorem is referenced by:  tltnle  18480  odutos  32943  trleile  32946  toslat  48771