Theorem termccd 49640

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 49639	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
3	2	thinccd 49584	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2113  Catccat 17578  TermCatctermc 49633

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 2115  ax-9 2123  ax-ext 2705  ax-nul 5248

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 2537  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6445  df-fv 6497  df-ov 7358  df-thinc 49579  df-termc 49634

This theorem is referenced by:  termchomn0  49645  funcsetc1ocl  49657  funcsetc1o  49658  isinito2lem  49659  isinito3  49661  termcterm  49674  termcterm2  49675  termc2  49679  termcarweu  49689  diag1f1olem  49694  diag1f1o  49695  diag2f1olem  49697  diag2f1o  49698  diagffth  49699  diagciso  49700  diagcic  49701  termfucterm  49705  uobeqterm  49707  isinito4a  49709  setc1onsubc  49763  lmdran  49832