Theorem termccatd 50408

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
termccatd	⊢ (𝜑 → 𝐶 ∈ Cat)

Proof of Theorem termccatd

Step	Hyp	Ref	Expression
1		termcthind.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ TermCat)
2	1	termcthind 50407	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
3	2	thinccatd 50352	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Catccat 17755  TermCatctermc 50401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7417  df-thinc 50347  df-termc 50402

This theorem is used by:  termchomn0  50413  funcsetc1ocl  50425  funcsetc1o  50426  isinito2lem  50427  isinito3  50429  termcterm  50442  termcterm2  50443  termc2  50447  termcarweu  50457  diag1f1olem  50462  diag1f1o  50463  diag2f1olem  50465  diag2f1o  50466  diagffth  50467  diagciso  50468  diagcic  50469  termfucterm  50473  uobeqterm  50475  isinito4a  50477  setc1onsubc  50531  lmdran  50600