Theorem zeroorcl

Description: Reverse closure for a zero object: If a class has a zero object, the class is a category. (Contributed by AV, 4-Apr-2020)

	Ref	Expression
Assertion	zeroorcl	⊢ ( 𝑍 ∈ ( ZeroO ‘ 𝐶 ) → 𝐶 ∈ Cat )

Proof

Step	Hyp	Ref	Expression
1		df-zeroo	⊢ ZeroO = ( 𝑐 ∈ Cat ↦ ( ( InitO ‘ 𝑐 ) ∩ ( TermO ‘ 𝑐 ) ) )
2	1	mptrcl	⊢ ( 𝑍 ∈ ( ZeroO ‘ 𝐶 ) → 𝐶 ∈ Cat )