Theorem termccd 49838

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 49837	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
3	2	thinccd 49782	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2114  Catccat 17599  TermCatctermc 49831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-thinc 49777  df-termc 49832

This theorem is referenced by:  termchomn0  49843  funcsetc1ocl  49855  funcsetc1o  49856  isinito2lem  49857  isinito3  49859  termcterm  49872  termcterm2  49873  termc2  49877  termcarweu  49887  diag1f1olem  49892  diag1f1o  49893  diag2f1olem  49895  diag2f1o  49896  diagffth  49897  diagciso  49898  diagcic  49899  termfucterm  49903  uobeqterm  49905  isinito4a  49907  setc1onsubc  49961  lmdran  50030