Theorem termcthind 50508

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 50507	. 2 ⊢ (𝐶 ∈ TermCat → 𝐶 ∈ ThinCat)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐶 ∈ ThinCat)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ThinCatcthinc 50447  TermCatctermc 50502

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 2732

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 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6483  df-fv 6535  df-termc 50503

This theorem is used by:  termccatd  50509  termchommo  50515  termcid  50516  termchom  50518  oppctermco  50535  functermc  50538  fulltermc  50541  termcterm2  50544  fucterm  50572