Theorem termcthin 50390

Description: A terminal category is a thin category. (Contributed by Zhi Wang, 16-Oct-2025.)

Assertion

Ref	Expression
termcthin	⊢ (𝐶 ∈ TermCat → 𝐶 ∈ ThinCat)

Proof of Theorem termcthin

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2762	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
2	1	istermc 50387	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥(Base‘𝐶) = {𝑥}))
3	2	simplbi 502	1 ⊢ (𝐶 ∈ TermCat → 𝐶 ∈ ThinCat)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {csn 4587  ‘cfv 6537  Basecbs 17305  ThinCatcthinc 50330  TermCatctermc 50385

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 2147  ax-9 2155  ax-ext 2734

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-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-termc 50386

This theorem is used by:  termcthind  50391