Theorem termcthin 50106

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 2765	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
2	1	istermc 50103	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥(Base‘𝐶) = {𝑥}))
3	2	simplbi 501	1 ⊢ (𝐶 ∈ TermCat → 𝐶 ∈ ThinCat)

Syntax hints:   → wi 4   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {csn 4585  ‘cfv 6525  Basecbs 17259  ThinCatcthinc 50046  TermCatctermc 50101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-iota 6481  df-fv 6533  df-termc 50102

This theorem is referenced by:  termcthind  50107