Theorem tospos

Description: A Toset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018)

	Ref	Expression
Assertion	tospos	⊢ ( 𝐹 ∈ Toset → 𝐹 ∈ Poset )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝐹 ) = ( Base ‘ 𝐹 )
2		eqid	⊢ ( le ‘ 𝐹 ) = ( le ‘ 𝐹 )
3	1 2	istos	⊢ ( 𝐹 ∈ Toset ↔ ( 𝐹 ∈ Poset ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐹 ) ∀ 𝑦 ∈ ( Base ‘ 𝐹 ) ( 𝑥 ( le ‘ 𝐹 ) 𝑦 ∨ 𝑦 ( le ‘ 𝐹 ) 𝑥 ) ) )
4	3	simplbi	⊢ ( 𝐹 ∈ Toset → 𝐹 ∈ Poset )