Theorem discthing 50811

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 2759	. . . . 5 ⊢ ({𝐼} = if(𝑥 = 𝑦, {𝐼}, ∅) → (𝑖 ∈ {𝐼} ↔ 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
4	3	mobidv 2577	. . . 4 ⊢ ({𝐼} = if(𝑥 = 𝑦, {𝐼}, ∅) → (∃*𝑖 𝑖 ∈ {𝐼} ↔ ∃*𝑖 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
5		eleq2w2 2759	. . . . 5 ⊢ (∅ = if(𝑥 = 𝑦, {𝐼}, ∅) → (𝑖 ∈ ∅ ↔ 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
6	5	mobidv 2577	. . . 4 ⊢ (∅ = if(𝑥 = 𝑦, {𝐼}, ∅) → (∃*𝑖 𝑖 ∈ ∅ ↔ ∃*𝑖 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
7		eqid 2763	. . . . 5 ⊢ {𝐼} = {𝐼}
8		mosn 50139	. . . . 5 ⊢ ({𝐼} = {𝐼} → ∃*𝑖 𝑖 ∈ {𝐼})
9	7, 8	mp1i 14	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑥 = 𝑦) → ∃*𝑖 𝑖 ∈ {𝐼})
10		eqid 2763	. . . . 5 ⊢ ∅ = ∅
11		mo0 50140	. . . . 5 ⊢ (∅ = ∅ → ∃*𝑖 𝑖 ∈ ∅)
12	10, 11	mp1i 14	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ 𝑥 = 𝑦) → ∃*𝑖 𝑖 ∈ ∅)
13	4, 6, 9, 12	ifbothda 4524	. . 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 2850	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑖 ∈ (𝑥𝐻𝑦) ↔ 𝑖 ∈ if(𝑥 = 𝑦, {𝐼}, ∅)))
16	15	mobidv 2577	. . 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 50786	1 ⊢ (𝜑 → 𝐶 ∈ ThinCat)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃*wmo 2565  ∅c0 4281  ifcif 4485  {csn 4588  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  Hom chom 17471  Catccat 17870  ThinCatcthinc 50767

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6515  df-fv 6569  df-ov 7449  df-thinc 50768

This theorem is used by: (None)