![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iszeroo | Structured version Visualization version GIF version |
Description: The predicate "is a zero object" of a category. (Contributed by AV, 3-Apr-2020.) |
Ref | Expression |
---|---|
isinito.b | โข ๐ต = (Baseโ๐ถ) |
isinito.h | โข ๐ป = (Hom โ๐ถ) |
isinito.c | โข (๐ โ ๐ถ โ Cat) |
isinito.i | โข (๐ โ ๐ผ โ ๐ต) |
Ref | Expression |
---|---|
iszeroo | โข (๐ โ (๐ผ โ (ZeroOโ๐ถ) โ (๐ผ โ (InitOโ๐ถ) โง ๐ผ โ (TermOโ๐ถ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isinito.c | . . . 4 โข (๐ โ ๐ถ โ Cat) | |
2 | isinito.b | . . . 4 โข ๐ต = (Baseโ๐ถ) | |
3 | isinito.h | . . . 4 โข ๐ป = (Hom โ๐ถ) | |
4 | 1, 2, 3 | zerooval 17954 | . . 3 โข (๐ โ (ZeroOโ๐ถ) = ((InitOโ๐ถ) โฉ (TermOโ๐ถ))) |
5 | 4 | eleq2d 2813 | . 2 โข (๐ โ (๐ผ โ (ZeroOโ๐ถ) โ ๐ผ โ ((InitOโ๐ถ) โฉ (TermOโ๐ถ)))) |
6 | elin 3959 | . 2 โข (๐ผ โ ((InitOโ๐ถ) โฉ (TermOโ๐ถ)) โ (๐ผ โ (InitOโ๐ถ) โง ๐ผ โ (TermOโ๐ถ))) | |
7 | 5, 6 | bitrdi 287 | 1 โข (๐ โ (๐ผ โ (ZeroOโ๐ถ) โ (๐ผ โ (InitOโ๐ถ) โง ๐ผ โ (TermOโ๐ถ)))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wb 205 โง wa 395 = wceq 1533 โ wcel 2098 โฉ cin 3942 โcfv 6536 Basecbs 17150 Hom chom 17214 Catccat 17614 InitOcinito 17940 TermOctermo 17941 ZeroOczeroo 17942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-zeroo 17945 |
This theorem is referenced by: iszeroi 17968 zrzeroorngc 20537 |
Copyright terms: Public domain | W3C validator |