Theorem termccd 49724

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 49723	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
3	2	thinccd 49668	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2113  Catccat 17587  TermCatctermc 49717

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 2708  ax-nul 5251

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 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361  df-thinc 49663  df-termc 49718

This theorem is referenced by:  termchomn0  49729  funcsetc1ocl  49741  funcsetc1o  49742  isinito2lem  49743  isinito3  49745  termcterm  49758  termcterm2  49759  termc2  49763  termcarweu  49773  diag1f1olem  49778  diag1f1o  49779  diag2f1olem  49781  diag2f1o  49782  diagffth  49783  diagciso  49784  diagcic  49785  termfucterm  49789  uobeqterm  49791  isinito4a  49793  setc1onsubc  49847  lmdran  49916