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Metamath Proof Explorer

Theorem ltrelnq

Description: Positive fraction 'less than' is a relation on positive fractions. (Contributed by NM, 14-Feb-1996) (Revised by Mario Carneiro, 27-Apr-2013) (New usage is discouraged.)

		Ref	Expression
	Assertion	ltrelnq	$⊢ <_{𝑸} \subseteq 𝑸 \times 𝑸$

Proof

Step	Hyp	Ref	Expression
1		df-ltnq	$⊢ <_{𝑸} = <_{𝑝𝑸} \cap (𝑸 \times 𝑸)$
2		inss2	$⊢ <_{𝑝𝑸} \cap (𝑸 \times 𝑸) \subseteq 𝑸 \times 𝑸$
3	1 2	eqsstri	$⊢ <_{𝑸} \subseteq 𝑸 \times 𝑸$