Theorem psrel

Description: A poset is a relation. (Contributed by NM, 12-May-2008)

	Ref	Expression
Assertion	psrel	⊢ ( 𝐴 ∈ PosetRel → Rel 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		isps	⊢ ( 𝐴 ∈ PosetRel → ( 𝐴 ∈ PosetRel ↔ ( Rel 𝐴 ∧ ( 𝐴 ∘ 𝐴 ) ⊆ 𝐴 ∧ ( 𝐴 ∩ ◡ 𝐴 ) = ( I ↾ ∪ ∪ 𝐴 ) ) ) )
2	1	ibi	⊢ ( 𝐴 ∈ PosetRel → ( Rel 𝐴 ∧ ( 𝐴 ∘ 𝐴 ) ⊆ 𝐴 ∧ ( 𝐴 ∩ ◡ 𝐴 ) = ( I ↾ ∪ ∪ 𝐴 ) ) )
3	2	simp1d	⊢ ( 𝐴 ∈ PosetRel → Rel 𝐴 )