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Metamath Proof Explorer

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