Theorem discthing 50271

Description: A discrete category, i.e., a category where all morphisms are identity morphisms, is thin. Example 3.26(1) of [Adamek] p. 33. (Contributed by Zhi Wang, 11-Nov-2025.)

Hypotheses

Ref	Expression
indcthing.b	⊢ (𝜑 → 𝐵 = (Base‘𝐶))
indcthing.h	⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
indcthing.c	⊢ (𝜑 → 𝐶 ∈ Cat)
discthing.i	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐻𝑦) = if(𝑥 = 𝑦, {𝐼}, ∅))

Assertion

Ref	Expression
discthing	⊢ (𝜑 → 𝐶 ∈ ThinCat)

Distinct variable groups:   𝑦,𝐵   𝑥,𝐶,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑥)   𝐻(𝑥, 𝑦)   𝐼(𝑥, 𝑦)

Proof of Theorem discthing

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		indcthing.b	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐶))
2		indcthing.h	. 2 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶))
3		eleq2w2 2762	. . . . 5 ⊢ ({𝐼} = if(𝑥 = 𝑦, {𝐼}, ∅) → (𝑖 ∈ {𝐼} ↔ 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
4	3	mobidv 2580	. . . 4 ⊢ ({𝐼} = if(𝑥 = 𝑦, {𝐼}, ∅) → (∃*𝑖 𝑖 ∈ {𝐼} ↔ ∃*𝑖 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
5		eleq2w2 2762	. . . . 5 ⊢ (∅ = if(𝑥 = 𝑦, {𝐼}, ∅) → (𝑖 ∈ ∅ ↔ 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
6	5	mobidv 2580	. . . 4 ⊢ (∅ = if(𝑥 = 𝑦, {𝐼}, ∅) → (∃*𝑖 𝑖 ∈ ∅ ↔ ∃*𝑖 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
7		eqid 2766	. . . . 5 ⊢ {𝐼} = {𝐼}
8		mosn 49623	. . . . 5 ⊢ ({𝐼} = {𝐼} → ∃*𝑖 𝑖 ∈ {𝐼})
9	7, 8	mp1i 14	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑥 = 𝑦) → ∃*𝑖 𝑖 ∈ {𝐼})
10		eqid 2766	. . . . 5 ⊢ ∅ = ∅
11		mo0 49624	. . . . 5 ⊢ (∅ = ∅ → ∃*𝑖 𝑖 ∈ ∅)
12	10, 11	mp1i 14	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ 𝑥 = 𝑦) → ∃*𝑖 𝑖 ∈ ∅)
13	4, 6, 9, 12	ifbothda 4529	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑖 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅))
14		discthing.i	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐻𝑦) = if(𝑥 = 𝑦, {𝐼}, ∅))
15	14	eleq2d 2852	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ (𝑥𝐻𝑦) ↔ 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
16	15	mobidv 2580	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃*𝑖 𝑖 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑖 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
17	13, 16	mpbird 260	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑖 𝑖 ∈ (𝑥𝐻𝑦))
18		indcthing.c	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
19	1, 2, 17, 18	isthincd 50246	1 ⊢ (𝜑 → 𝐶 ∈ ThinCat)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃*wmo 2568  ∅c0 4289  ifcif 4490  {csn 4592  ‘cfv 6540  (class class class)co 7416  Basecbs 17279  Hom chom 17331  Catccat 17730  ThinCatcthinc 50227

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5260  ax-nul 5272

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4876  df-br 5113  df-iota 6496  df-fv 6548  df-ov 7419  df-thinc 50228

This theorem is used by: (None)