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Metamath Proof Explorer

Theorem 0nnq

Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995) (Revised by Mario Carneiro, 27-Apr-2013) (New usage is discouraged.)

		Ref	Expression
	Assertion	0nnq	⊢ ¬ ∅ ∈ Q

Proof

Step	Hyp	Ref	Expression
1		0nelxp	⊢ ¬ ∅ ∈ ( N × N )
2		df-nq	⊢ Q = { 𝑦 ∈ ( N × N ) ∣ ∀ 𝑥 ∈ ( N × N ) ( 𝑦 ~_Q 𝑥 → ¬ ( 2^nd ‘ 𝑥 ) <_N ( 2^nd ‘ 𝑦 ) ) }
3	2	ssrab3	⊢ Q ⊆ ( N × N )
4	3	sseli	⊢ ( ∅ ∈ Q → ∅ ∈ ( N × N ) )
5	1 4	mto	⊢ ¬ ∅ ∈ Q