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	$$⊢{<}_{𝑸}={<}_{\mathrm{𝑝𝑸}}\cap \left(𝑸\times 𝑸\right)$$
2		inss2	$$⊢{<}_{\mathrm{𝑝𝑸}}\cap \left(𝑸\times 𝑸\right)\subseteq 𝑸\times 𝑸$$
3	1 2	eqsstri	$$⊢{<}_{𝑸}\subseteq 𝑸\times 𝑸$$