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Theorem catprs2 48874
Description: A category equipped with the induced preorder, where an object 𝑥 is defined to be "less than or equal to" 𝑦 iff there is a morphism from 𝑥 to 𝑦, is a preordered set, or a proset. The category might not be thin. See catprsc 48875 and catprsc2 48876 for constructions satisfying the hypothesis "catprs.1". See catprs 48873 for a more primitive version. See prsthinc 49084 for constructing a thin category from a proset. (Contributed by Zhi Wang, 18-Sep-2024.)
Hypotheses
Ref Expression
catprs.1 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))
catprs.b (𝜑𝐵 = (Base‘𝐶))
catprs.h (𝜑𝐻 = (Hom ‘𝐶))
catprs.c (𝜑𝐶 ∈ Cat)
catprs2.l (𝜑 = (le‘𝐶))
Assertion
Ref Expression
catprs2 (𝜑𝐶 ∈ Proset )
Distinct variable groups:   𝑥, ,𝑦   𝑥,𝐵,𝑦   𝑥,𝐻,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem catprs2
Dummy variables 𝑤 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catprs.1 . . . 4 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))
2 catprs.b . . . 4 (𝜑𝐵 = (Base‘𝐶))
3 catprs.h . . . 4 (𝜑𝐻 = (Hom ‘𝐶))
4 catprs.c . . . 4 (𝜑𝐶 ∈ Cat)
51, 2, 3, 4catprs 48873 . . 3 ((𝜑 ∧ (𝑤𝐵𝑣𝐵𝑢𝐵)) → (𝑤 𝑤 ∧ ((𝑤 𝑣𝑣 𝑢) → 𝑤 𝑢)))
65ralrimivvva 3204 . 2 (𝜑 → ∀𝑤𝐵𝑣𝐵𝑢𝐵 (𝑤 𝑤 ∧ ((𝑤 𝑣𝑣 𝑢) → 𝑤 𝑢)))
7 catprs2.l . . 3 (𝜑 = (le‘𝐶))
82, 7, 4isprsd 48825 . 2 (𝜑 → (𝐶 ∈ Proset ↔ ∀𝑤𝐵𝑣𝐵𝑢𝐵 (𝑤 𝑤 ∧ ((𝑤 𝑣𝑣 𝑢) → 𝑤 𝑢))))
96, 8mpbird 257 1 (𝜑𝐶 ∈ Proset )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2939  wral 3060  c0 4332   class class class wbr 5141  cfv 6559  (class class class)co 7429  Basecbs 17243  lecple 17300  Hom chom 17304  Catccat 17703   Proset cproset 18334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-riota 7386  df-ov 7432  df-cat 17707  df-cid 17708  df-proset 18336
This theorem is referenced by: (None)
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