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Mirrors > Home > MPE Home > Th. List > zeroorcl | Structured version Visualization version GIF version |
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 |
---|---|
zeroorcl | ⊢ (𝑍 ∈ (ZeroO‘𝐶) → 𝐶 ∈ Cat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-zeroo 16995 | . 2 ⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐))) | |
2 | 1 | mptrcl 6536 | 1 ⊢ (𝑍 ∈ (ZeroO‘𝐶) → 𝐶 ∈ Cat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2164 ∩ cin 3797 ‘cfv 6123 Catccat 16677 InitOcinito 16990 TermOctermo 16991 ZeroOczeroo 16992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-xp 5348 df-rel 5349 df-cnv 5350 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fv 6131 df-zeroo 16995 |
This theorem is referenced by: (None) |
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