Theorem termcthind 49589

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 49588	. 2 ⊢ (𝐶 ∈ TermCat → 𝐶 ∈ ThinCat)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐶 ∈ ThinCat)

Syntax hints:   → wi 4   ∈ wcel 2111  ThinCatcthinc 49528  TermCatctermc 49583

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 2113  ax-9 2121  ax-ext 2703

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 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-termc 49584

This theorem is referenced by:  termccd  49590  termchommo  49596  termcid  49597  termchom  49599  oppctermco  49616  functermc  49619  fulltermc  49622  termcterm2  49625  fucterm  49653