Theorem termccd 49511

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 49510	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
3	2	thinccd 49455	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2111  Catccat 17565  TermCatctermc 49504

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  ax-nul 5239

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-mo 2535  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-iota 6432  df-fv 6484  df-ov 7344  df-thinc 49450  df-termc 49505

This theorem is referenced by:  termchomn0  49516  funcsetc1ocl  49528  funcsetc1o  49529  isinito2lem  49530  isinito3  49532  termcterm  49545  termcterm2  49546  termc2  49550  termcarweu  49560  diag1f1olem  49565  diag1f1o  49566  diag2f1olem  49568  diag2f1o  49569  diagffth  49570  diagciso  49571  diagcic  49572  termfucterm  49576  uobeqterm  49578  isinito4a  49580  setc1onsubc  49634  lmdran  49703