Theorem eluzge3nn

Description: If an integer is greater than 3, then it is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018)

	Ref	Expression
Assertion	eluzge3nn	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) → 𝑁 ∈ ℕ )

Proof

Step	Hyp	Ref	Expression
1		1z	⊢ 1 ∈ ℤ
2		1le3	⊢ 1 ≤ 3
3		eluzuzle	⊢ ( ( 1 ∈ ℤ ∧ 1 ≤ 3 ) → ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) → 𝑁 ∈ ( ℤ≥ ‘ 1 ) ) )
4	1 2 3	mp2an	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) → 𝑁 ∈ ( ℤ≥ ‘ 1 ) )
5		elnnuz	⊢ ( 𝑁 ∈ ℕ ↔ 𝑁 ∈ ( ℤ≥ ‘ 1 ) )
6	4 5	sylibr	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) → 𝑁 ∈ ℕ )