MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  zeroorcl Structured version   Visualization version   GIF version

Theorem zeroorcl 17248
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.
StepHypRef Expression
1 df-zeroo 17245 . 2 ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
21mptrcl 6754 1 (𝑍 ∈ (ZeroO‘𝐶) → 𝐶 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2111  cin 3880  cfv 6324  Catccat 16927  InitOcinito 17240  TermOctermo 17241  ZeroOczeroo 17242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-xp 5525  df-rel 5526  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fv 6332  df-zeroo 17245
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator