Theorem nninf

Description: The infimum of the set of positive integers is one. (Contributed by NM, 16-Jun-2005) (Revised by AV, 5-Sep-2020)

	Ref	Expression
Assertion	nninf	⊢ inf ( ℕ , ℝ , < ) = 1

Proof

Step	Hyp	Ref	Expression
1		nnuz	⊢ ℕ = ( ℤ≥ ‘ 1 )
2	1	infeq1i	⊢ inf ( ℕ , ℝ , < ) = inf ( ( ℤ≥ ‘ 1 ) , ℝ , < )
3		1z	⊢ 1 ∈ ℤ
4	3	uzinfi	⊢ inf ( ( ℤ≥ ‘ 1 ) , ℝ , < ) = 1
5	2 4	eqtri	⊢ inf ( ℕ , ℝ , < ) = 1