Theorem archd

Description: Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of Apostol p. 26. (Contributed by Glauco Siliprandi, 1-Feb-2025)

	Ref	Expression
Hypothesis	archd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
Assertion	archd	⊢ ( 𝜑 → ∃ 𝑛 ∈ ℕ 𝐴 < 𝑛 )

Proof

Step	Hyp	Ref	Expression
1		archd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		arch	⊢ ( 𝐴 ∈ ℝ → ∃ 𝑛 ∈ ℕ 𝐴 < 𝑛 )
3	1 2	syl	⊢ ( 𝜑 → ∃ 𝑛 ∈ ℕ 𝐴 < 𝑛 )