Theorem termcthind 49744

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 49743	. 2 ⊢ (𝐶 ∈ TermCat → 𝐶 ∈ ThinCat)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐶 ∈ ThinCat)

Syntax hints:   → wi 4   ∈ wcel 2113  ThinCatcthinc 49683  TermCatctermc 49738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-termc 49739

This theorem is referenced by:  termccd  49745  termchommo  49751  termcid  49752  termchom  49754  oppctermco  49771  functermc  49774  fulltermc  49777  termcterm2  49780  fucterm  49808