Theorem termccd 49969

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 49968	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
3	2	thinccd 49913	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2119  Catccat 17621  TermCatctermc 49962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-ov 7359  df-thinc 49908  df-termc 49963

This theorem is referenced by:  termchomn0  49974  funcsetc1ocl  49986  funcsetc1o  49987  isinito2lem  49988  isinito3  49990  termcterm  50003  termcterm2  50004  termc2  50008  termcarweu  50018  diag1f1olem  50023  diag1f1o  50024  diag2f1olem  50026  diag2f1o  50027  diagffth  50028  diagciso  50029  diagcic  50030  termfucterm  50034  uobeqterm  50036  isinito4a  50038  setc1onsubc  50092  lmdran  50161