Theorem indcthing 49935

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 2736	. . . 4 ⊢ {𝐹} = {𝐹}
4		mosn 49288	. . . 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 2822	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ {𝐹}))
8	7	mobidv 2549	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ {𝐹}))
9	5, 8	mpbiri 258	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
10		indcthing.c	. 2 ⊢ (𝜑 → 𝐶 ∈ Cat)
11	1, 2, 9, 10	isthincd 49911	1 ⊢ (𝜑 → 𝐶 ∈ ThinCat)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃*wmo 2537  {csn 4567  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  Hom chom 17231  Catccat 17630  ThinCatcthinc 49892

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-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-nul 5241

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-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-thinc 49893

This theorem is referenced by: (None)