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Metamath Proof Explorer

Theorem nn0ssz

Description: Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004)

		Ref	Expression
	Assertion	nn0ssz	$⊢ ℕ_{0} \subseteq ℤ$

Step	Hyp	Ref	Expression
1		df-n0	$⊢ ℕ_{0} = ℕ \cup \{0\}$
2		nnssz	$⊢ ℕ \subseteq ℤ$
3		0z	$⊢ 0 \in ℤ$
4		c0ex	$⊢ 0 \in V$
5	4	snss	$⊢ 0 \in ℤ \leftrightarrow \{0\} \subseteq ℤ$
6	3 5	mpbi	$⊢ \{0\} \subseteq ℤ$
7	2 6	unssi	$⊢ ℕ \cup \{0\} \subseteq ℤ$
8	1 7	eqsstri	$⊢ ℕ_{0} \subseteq ℤ$