Theorem termccd 50061

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 50060	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
3	2	thinccd 50005	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2141  Catccat 17687  TermCatctermc 50054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5253

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6472  df-fv 6524  df-ov 7394  df-thinc 50000  df-termc 50055

This theorem is referenced by:  termchomn0  50066  funcsetc1ocl  50078  funcsetc1o  50079  isinito2lem  50080  isinito3  50082  termcterm  50095  termcterm2  50096  termc2  50100  termcarweu  50110  diag1f1olem  50115  diag1f1o  50116  diag2f1olem  50118  diag2f1o  50119  diagffth  50120  diagciso  50121  diagcic  50122  termfucterm  50126  uobeqterm  50128  isinito4a  50130  setc1onsubc  50184  lmdran  50253