Theorem zeroorcl 18067

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.)

Assertion

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

Proof of Theorem zeroorcl

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-zeroo 18061	. 2 ⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
2	1	mptrcl 7003	1 ⊢ (𝑍 ∈ (ZeroO‘𝐶) → 𝐶 ∈ Cat)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146   ∩ cin 3905  ‘cfv 6540  Catccat 17738  InitOcinito 18056  TermOctermo 18057  ZeroOczeroo 18058

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fv 6548  df-zeroo 18061

This theorem is used by:  zeroo2  50045  oppczeroo  50048