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Metamath Proof Explorer

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 ) )