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Metamath Proof Explorer

Theorem psrel

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

		Ref	Expression
	Assertion	psrel	$⊢ A \in PosetRel \to Rel A$

Step	Hyp	Ref	Expression
1		isps	$⊢ A \in PosetRel \to (A \in PosetRel \leftrightarrow (Rel A \land A \circ A \subseteq A \land A \cap A^{-1} = {I ↾}_{⋃ ⋃ A}))$
2	1	ibi	$⊢ A \in PosetRel \to (Rel A \land A \circ A \subseteq A \land A \cap A^{-1} = {I ↾}_{⋃ ⋃ A})$
3	2	simp1d	$⊢ A \in PosetRel \to Rel A$