Theorem nnuz

Description: Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005)

	Ref	Expression
Assertion	nnuz	⊢ ℕ = ( ℤ≥ ‘ 1 )

Proof

Step	Hyp	Ref	Expression
1		nnzrab	⊢ ℕ = { 𝑘 ∈ ℤ ∣ 1 ≤ 𝑘 }
2		1z	⊢ 1 ∈ ℤ
3		uzval	⊢ ( 1 ∈ ℤ → ( ℤ≥ ‘ 1 ) = { 𝑘 ∈ ℤ ∣ 1 ≤ 𝑘 } )
4	2 3	ax-mp	⊢ ( ℤ≥ ‘ 1 ) = { 𝑘 ∈ ℤ ∣ 1 ≤ 𝑘 }
5	1 4	eqtr4i	⊢ ℕ = ( ℤ≥ ‘ 1 )