Theorem termccd 49954

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 49953	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
3	2	thinccd 49898	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2114  Catccat 17630  TermCatctermc 49947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-thinc 49893  df-termc 49948

This theorem is referenced by:  termchomn0  49959  funcsetc1ocl  49971  funcsetc1o  49972  isinito2lem  49973  isinito3  49975  termcterm  49988  termcterm2  49989  termc2  49993  termcarweu  50003  diag1f1olem  50008  diag1f1o  50009  diag2f1olem  50011  diag2f1o  50012  diagffth  50013  diagciso  50014  diagcic  50015  termfucterm  50019  uobeqterm  50021  isinito4a  50023  setc1onsubc  50077  lmdran  50146