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Metamath Proof Explorer

Theorem nn0sscn

Description: Nonnegative integers are a subset of the complex numbers. (Contributed by NM, 9-May-2004) Reduce dependencies on axioms. (Revised by Steven Nguyen, 8-Oct-2022)

		Ref	Expression
	Assertion	nn0sscn	$⊢ ℕ_{0} \subseteq ℂ$

Proof

Step	Hyp	Ref	Expression
1		df-n0	$⊢ ℕ_{0} = ℕ \cup \{0\}$
2		nnsscn	$⊢ ℕ \subseteq ℂ$
3		0cn	$⊢ 0 \in ℂ$
4		snssi	$⊢ 0 \in ℂ \to \{0\} \subseteq ℂ$
5	3 4	ax-mp	$⊢ \{0\} \subseteq ℂ$
6	2 5	unssi	$⊢ ℕ \cup \{0\} \subseteq ℂ$
7	1 6	eqsstri	$⊢ ℕ_{0} \subseteq ℂ$