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 ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 𝑌𝑌 𝑋 ) )

Proof

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