Theorem indcthing 50688

Description: An indiscrete category, i.e., a category where all hom-sets have exactly one morphism, is thin. (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)
indcthing.i	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐻𝑦) = {𝐹})

Assertion

Ref	Expression
indcthing	⊢ (𝜑 → 𝐶 ∈ ThinCat)

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

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

Proof of Theorem indcthing

Dummy variable 𝑓 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		eqid 2760	. . . 4 ⊢ {𝐹} = {𝐹}
4		mosn 50017	. . . 4 ⊢ ({𝐹} = {𝐹} → ∃*𝑓 𝑓 ∈ {𝐹})
5	3, 4	ax-mp 5	. . 3 ⊢ ∃*𝑓 𝑓 ∈ {𝐹}
6		indcthing.i	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐻𝑦) = {𝐹})
7	6	eleq2d 2846	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ {𝐹}))
8	7	mobidv 2574	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ {𝐹}))
9	5, 8	mpbiri 261	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
10		indcthing.c	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
11	1, 2, 9, 10	isthincd 50664	1 ⊢ (𝜑 → 𝐶 ∈ ThinCat)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃*wmo 2562  {csn 4584  ‘cfv 6538  (class class class)co 7420  Basecbs 17392  Hom chom 17444  Catccat 17843  ThinCatcthinc 50645

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-ov 7423  df-thinc 50646

This theorem is used by: (None)