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Metamath Proof Explorer

Theorem 0npi

Description: The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995) (New usage is discouraged.)

		Ref	Expression
	Assertion	0npi	⊢ ¬ ∅ ∈ N

Step	Hyp	Ref	Expression
1		eqid	⊢ ∅ = ∅
2		elni	⊢ ( ∅ ∈ N ↔ ( ∅ ∈ ω ∧ ∅ ≠ ∅ ) )
3	2	simprbi	⊢ ( ∅ ∈ N → ∅ ≠ ∅ )
4	3	necon2bi	⊢ ( ∅ = ∅ → ¬ ∅ ∈ N )
5	1 4	ax-mp	⊢ ¬ ∅ ∈ N