Theorem termccd 50405

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 50404	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
3	2	thinccd 50349	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Catccat 17752  TermCatctermc 50398

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 7416  df-thinc 50344  df-termc 50399

This theorem is used by:  termchomn0  50410  funcsetc1ocl  50422  funcsetc1o  50423  isinito2lem  50424  isinito3  50426  termcterm  50439  termcterm2  50440  termc2  50444  termcarweu  50454  diag1f1olem  50459  diag1f1o  50460  diag2f1olem  50462  diag2f1o  50463  diagffth  50464  diagciso  50465  diagcic  50466  termfucterm  50470  uobeqterm  50472  isinito4a  50474  setc1onsubc  50528  lmdran  50597