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Theorem 0thincg 48851
Description: Any structure with an empty set of objects is a thin category. (Contributed by Zhi Wang, 17-Sep-2024.)
Assertion
Ref Expression
0thincg ((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ ThinCat)

Proof of Theorem 0thincg
Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0catg 17733 . 2 ((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat)
2 ral0 4519 . . . 4 𝑥 ∈ ∅ ∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)
3 raleq 3321 . . . 4 (∅ = (Base‘𝐶) → (∀𝑥 ∈ ∅ ∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
42, 3mpbii 233 . . 3 (∅ = (Base‘𝐶) → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
54adantl 481 . 2 ((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
6 eqid 2735 . . 3 (Base‘𝐶) = (Base‘𝐶)
7 eqid 2735 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
86, 7isthinc 48821 . 2 (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
91, 5, 8sylanbrc 583 1 ((𝐶𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ ThinCat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  ∃*wmo 2536  wral 3059  c0 4339  cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  Catccat 17709  ThinCatcthinc 48819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-cat 17713  df-thinc 48820
This theorem is referenced by:  0thinc  48852
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