Theorem elpqn

Description: Each positive fraction is an ordered pair of positive integers (the numerator and denominator, in "lowest terms". (Contributed by Mario Carneiro, 28-Apr-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	elpqn	⊢ ( 𝐴 ∈ Q → 𝐴 ∈ ( N × N ) )

Proof

Step	Hyp	Ref	Expression
1		df-nq	⊢ Q = { 𝑦 ∈ ( N × N ) ∣ ∀ 𝑥 ∈ ( N × N ) ( 𝑦 ~Q 𝑥 → ¬ ( 2nd ‘ 𝑥 ) <N ( 2nd ‘ 𝑦 ) ) }
2	1	ssrab3	⊢ Q ⊆ ( N × N )
3	2	sseli	⊢ ( 𝐴 ∈ Q → 𝐴 ∈ ( N × N ) )