Theorem termcthind 50818

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 50817	. 2 ⊢ (𝐶 ∈ TermCat → 𝐶 ∈ ThinCat)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐶 ∈ ThinCat)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  ThinCatcthinc 50757  TermCatctermc 50812

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6512  df-fv 6566  df-termc 50813

This theorem is used by:  termccatd  50819  termchommo  50825  termcid  50826  termchom  50828  oppctermco  50845  functermc  50848  fulltermc  50851  termcterm2  50854  fucterm  50885