Theorem nnssre

Description: The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997) (Revised by Mario Carneiro, 16-Jun-2013)

	Ref	Expression
Assertion	nnssre	⊢ ℕ ⊆ ℝ

Proof

Step	Hyp	Ref	Expression
1		1re	⊢ 1 ∈ ℝ
2		peano2re	⊢ ( 𝑥 ∈ ℝ → ( 𝑥 + 1 ) ∈ ℝ )
3	2	rgen	⊢ ∀ 𝑥 ∈ ℝ ( 𝑥 + 1 ) ∈ ℝ
4		peano5nni	⊢ ( ( 1 ∈ ℝ ∧ ∀ 𝑥 ∈ ℝ ( 𝑥 + 1 ) ∈ ℝ ) → ℕ ⊆ ℝ )
5	1 3 4	mp2an	⊢ ℕ ⊆ ℝ