Theorem termcthind 50629

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

Hypothesis

Ref	Expression
termcthind.c	⊢ (𝜑 → 𝐶 ∈ TermCat)

Assertion

Ref	Expression
termcthind	⊢ (𝜑 → 𝐶 ∈ ThinCat)

Proof of Theorem termcthind

Step	Hyp	Ref	Expression
1		termcthind.c	. 2 ⊢ (𝜑 → 𝐶 ∈ TermCat)
2		termcthin 50628	. 2 ⊢ (𝐶 ∈ TermCat → 𝐶 ∈ ThinCat)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐶 ∈ ThinCat)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  ThinCatcthinc 50568  TermCatctermc 50623

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-ext 2732

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-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-termc 50624

This theorem is used by:  termccatd  50630  termchommo  50636  termcid  50637  termchom  50639  oppctermco  50656  functermc  50659  fulltermc  50662  termcterm2  50665  fucterm  50693