Theorem zeroorcl 17894

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 17888	. 2 ⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
2	1	mptrcl 6933	1 ⊢ (𝑍 ∈ (ZeroO‘𝐶) → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2111   ∩ cin 3896  ‘cfv 6476  Catccat 17565  InitOcinito 17883  TermOctermo 17884  ZeroOczeroo 17885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-xp 5617  df-rel 5618  df-cnv 5619  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fv 6484  df-zeroo 17888

This theorem is referenced by:  zeroo2  49266  oppczeroo  49269