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| Mirrors > Home > MPE Home > Th. List > Mathboxes > 0thincg | Structured version Visualization version GIF version | ||
| Description: Any structure with an empty set of objects is a thin category. (Contributed by Zhi Wang, 17-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| 0thincg | ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ ThinCat) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0catg 17731 | . 2 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ Cat) | |
| 2 | ral0 4513 | . . . 4 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) | |
| 3 | raleq 3323 | . . . 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 49069 | . 2 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) | 
| 9 | 1, 5, 8 | sylanbrc 583 | 1 ⊢ ((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ ThinCat) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃*wmo 2538 ∀wral 3061 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Hom chom 17308 Catccat 17707 ThinCatcthinc 49067 | 
| 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-ext 2708 ax-nul 5306 | 
| 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-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-cat 17711 df-thinc 49068 | 
| This theorem is referenced by: 0thinc 49108 | 
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