Theorem termccd 49484

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

Hypothesis

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

Assertion

Ref	Expression
termccd	⊢ (𝜑 → 𝐶 ∈ Cat)

Proof of Theorem termccd

Step	Hyp	Ref	Expression
1		termcthind.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ TermCat)
2	1	termcthind 49483	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
3	2	thinccd 49428	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2109  Catccat 17589  TermCatctermc 49477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5248

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-iota 6442  df-fv 6494  df-ov 7356  df-thinc 49423  df-termc 49478

This theorem is referenced by:  termchomn0  49489  funcsetc1ocl  49501  funcsetc1o  49502  isinito2lem  49503  isinito3  49505  termcterm  49518  termcterm2  49519  termc2  49523  termcarweu  49533  diag1f1olem  49538  diag1f1o  49539  diag2f1olem  49541  diag2f1o  49542  diagffth  49543  diagciso  49544  diagcic  49545  termfucterm  49549  uobeqterm  49551  isinito4a  49553  setc1onsubc  49607  lmdran  49676