Theorem termccd 49468

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 49467	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
3	2	thinccd 49412	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2109  Catccat 17625  TermCatctermc 49461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-thinc 49407  df-termc 49462

This theorem is referenced by:  termchomn0  49473  funcsetc1ocl  49485  funcsetc1o  49486  isinito2lem  49487  isinito3  49489  termcterm  49502  termcterm2  49503  termc2  49507  termcarweu  49517  diag1f1olem  49522  diag1f1o  49523  diag2f1olem  49525  diag2f1o  49526  diagffth  49527  diagciso  49528  diagcic  49529  termfucterm  49533  uobeqterm  49535  isinito4a  49537  setc1onsubc  49591  lmdran  49660