Theorem zeroorcl 18174

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 18168	. 2 ⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐)))
2	1	mptrcl 7006	1 ⊢ (𝑍 ∈ (ZeroO‘𝐶) → 𝐶 ∈ Cat)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144   ∩ cin 4088  ‘cfv 6541  Catccat 17845  InitOcinito 18163  TermOctermo 18164  ZeroOczeroo 18165

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fv 6549  df-zeroo 18168

This theorem is used by:  zeroo2  50463  oppczeroo  50466