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Metamath Proof Explorer

Theorem nnsscn

Description: The positive integers are a subset of the complex numbers. Remark: this could also be proven from nnssre and ax-resscn at the cost of using more axioms. (Contributed by NM, 2-Aug-2004) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022)

		Ref	Expression
	Assertion	nnsscn	⊢ ℕ ⊆ ℂ

Proof

Step	Hyp	Ref	Expression
1		ax-1cn	⊢ 1 ∈ ℂ
2		peano2cn	⊢ ( 𝑥 ∈ ℂ → ( 𝑥 + 1 ) ∈ ℂ )
3	2	rgen	⊢ ∀ 𝑥 ∈ ℂ ( 𝑥 + 1 ) ∈ ℂ
4		peano5nni	⊢ ( ( 1 ∈ ℂ ∧ ∀ 𝑥 ∈ ℂ ( 𝑥 + 1 ) ∈ ℂ ) → ℕ ⊆ ℂ )
5	1 3 4	mp2an	⊢ ℕ ⊆ ℂ