Theorem termccd 49441

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 49440	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
3	2	thinccd 49385	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2109  Catccat 17601  TermCatctermc 49434

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 5256

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 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-iota 6452  df-fv 6507  df-ov 7372  df-thinc 49380  df-termc 49435

This theorem is referenced by:  termchomn0  49446  funcsetc1ocl  49458  funcsetc1o  49459  isinito2lem  49460  isinito3  49462  termcterm  49475  termcterm2  49476  termc2  49480  termcarweu  49490  diag1f1olem  49495  diag1f1o  49496  diag2f1olem  49498  diag2f1o  49499  diagffth  49500  diagciso  49501  diagcic  49502  termfucterm  49506  uobeqterm  49508  isinito4a  49510  setc1onsubc  49564  lmdran  49633