Theorem termccatd 50586

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 50585	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
3	2	thinccatd 50530	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Catccat 17838  TermCatctermc 50579

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 2733  ax-nul 5260

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 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  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 6494  df-fv 6546  df-ov 7423  df-thinc 50525  df-termc 50580

This theorem is used by:  termchomn0  50591  funcsetc1ocl  50603  funcsetc1o  50604  isinito2lem  50605  isinito3  50607  termcterm  50620  termcterm2  50621  termc2  50625  termcarweu  50635  diag1f1olem  50640  diag1f1o  50641  diag2f1olem  50643  diag2f1o  50644  diagffth  50645  diagciso  50646  diagcic  50647  termfucterm  50651  uobeqterm  50653  isinito4a  50655  setc1onsubc  50709  lmdran  50778