Theorem 0z

Description: Zero is an integer. (Contributed by NM, 12-Jan-2002)

	Ref	Expression
Assertion	0z	⊢ 0 ∈ ℤ

Proof

Step	Hyp	Ref	Expression
1		0re	⊢ 0 ∈ ℝ
2		eqid	⊢ 0 = 0
3	2	3mix1i	⊢ ( 0 = 0 ∨ 0 ∈ ℕ ∨ - 0 ∈ ℕ )
4		elz	⊢ ( 0 ∈ ℤ ↔ ( 0 ∈ ℝ ∧ ( 0 = 0 ∨ 0 ∈ ℕ ∨ - 0 ∈ ℕ ) ) )
5	1 3 4	mpbir2an	⊢ 0 ∈ ℤ