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Metamath Proof Explorer

Theorem tleile

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

Ref Expression
Hypotheses tleile.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
tleile.l ⊢ ≤ = ( le ‘ 𝐾 )
Assertion tleile ( ( 𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋 ) )

Proof

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