Theorem 0thincg 49817

Description: Any structure with an empty set of objects is a thin category. (Contributed by Zhi Wang, 17-Sep-2024.)

Assertion

Ref	Expression
0thincg	⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ ThinCat)

Proof of Theorem 0thincg

Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0catg 17623	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat)
2		ral0 4453	. . . 4 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)
3		raleq 3295	. . . 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)))
4	2, 3	mpbii 233	. . 3 ⊢ (∅ = (Base‘𝐶) → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
5	4	adantl 481	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
6		eqid 2737	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
7		eqid 2737	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8	6, 7	isthinc 49778	. 2 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)))
9	1, 5, 8	sylanbrc 584	1 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ ThinCat)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃*wmo 2538  ∀wral 3052  ∅c0 4287  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  Hom chom 17200  Catccat 17599  ThinCatcthinc 49776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-cat 17603  df-thinc 49777

This theorem is referenced by:  0thinc  49818