Theorem termccd 50137

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 50136	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
3	2	thinccd 50081	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2149  Catccat 17716  TermCatctermc 50130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5268

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-iota 6490  df-fv 6542  df-ov 7411  df-thinc 50076  df-termc 50131

This theorem is referenced by:  termchomn0  50142  funcsetc1ocl  50154  funcsetc1o  50155  isinito2lem  50156  isinito3  50158  termcterm  50171  termcterm2  50172  termc2  50176  termcarweu  50186  diag1f1olem  50191  diag1f1o  50192  diag2f1olem  50194  diag2f1o  50195  diagffth  50196  diagciso  50197  diagcic  50198  termfucterm  50202  uobeqterm  50204  isinito4a  50206  setc1onsubc  50260  lmdran  50329