Theorem termccd 50257

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 50256	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
3	2	thinccd 50201	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2143  Catccat 17715  TermCatctermc 50250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-thinc 50196  df-termc 50251

This theorem is referenced by:  termchomn0  50262  funcsetc1ocl  50274  funcsetc1o  50275  isinito2lem  50276  isinito3  50278  termcterm  50291  termcterm2  50292  termc2  50296  termcarweu  50306  diag1f1olem  50311  diag1f1o  50312  diag2f1olem  50314  diag2f1o  50315  diagffth  50316  diagciso  50317  diagcic  50318  termfucterm  50322  uobeqterm  50324  isinito4a  50326  setc1onsubc  50380  lmdran  50449