Theorem tleile 18376

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 1138	. 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
2		simp3 1139	. 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 18373	. . . 4 ⊢ (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
6	5	simprbi 497	. . 3 ⊢ (𝐾 ∈ Toset → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
7	6	3ad2ant1 1134	. 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥))
8		breq1 5089	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
9		breq2 5090	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ 𝑋))
10	8, 9	orbi12d 919	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ (𝑋 ≤ 𝑦 ∨ 𝑦 ≤ 𝑋)))
11		breq2 5090	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
12		breq1 5089	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 ≤ 𝑋 ↔ 𝑌 ≤ 𝑋))
13	11, 12	orbi12d 919	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 ∨ 𝑦 ≤ 𝑋) ↔ (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋)))
14	10, 13	rspc2va 3577	. 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 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   class class class wbr 5086  ‘cfv 6492  Basecbs 17170  lecple 17218  Posetcpo 18264  Tosetctos 18371

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-ext 2709  ax-nul 5241

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-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6448  df-fv 6500  df-toset 18372

This theorem is referenced by:  tltnle  18377  odutos  33043  trleile  33046  toslat  49469