Theorem tsrps

Description: A toset is a poset. (Contributed by Mario Carneiro, 9-Sep-2015)

	Ref	Expression
Assertion	tsrps	⊢ ( 𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ dom 𝑅 = dom 𝑅
2	1	istsr	⊢ ( 𝑅 ∈ TosetRel ↔ ( 𝑅 ∈ PosetRel ∧ ( dom 𝑅 × dom 𝑅 ) ⊆ ( 𝑅 ∪ ◡ 𝑅 ) ) )
3	2	simplbi	⊢ ( 𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel )