Theorem termcthin 49836

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 2737	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
2	1	istermc 49833	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥(Base‘𝐶) = {𝑥}))
3	2	simplbi 496	1 ⊢ (𝐶 ∈ TermCat → 𝐶 ∈ ThinCat)

Syntax hints:   → wi 4   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {csn 4582  ‘cfv 6500  Basecbs 17148  ThinCatcthinc 49776  TermCatctermc 49831

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-ext 2709

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-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-termc 49832

This theorem is referenced by:  termcthind  49837