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Metamath Proof Explorer

Theorem nn0zrab

Description: Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004)

		Ref	Expression
	Assertion	nn0zrab	$⊢ ℕ_{0} = \{x \in ℤ \| 0 \leq x\}$

Step	Hyp	Ref	Expression
1		elnn0z	$⊢ x \in ℕ_{0} \leftrightarrow (x \in ℤ \land 0 \leq x)$
2	1	eqabi	$⊢ ℕ_{0} = \{x \| (x \in ℤ \land 0 \leq x)\}$
3		df-rab	$⊢ \{x \in ℤ \| 0 \leq x\} = \{x \| (x \in ℤ \land 0 \leq x)\}$
4	2 3	eqtr4i	$⊢ ℕ_{0} = \{x \in ℤ \| 0 \leq x\}$