Theorem thinccd 49309

Description: A thin category is a category (deduction form). (Contributed by Zhi Wang, 24-Sep-2024.)

Hypothesis

Ref	Expression
thinccd.c	⊢ (𝜑 → 𝐶 ∈ ThinCat)

Assertion

Ref	Expression
thinccd	⊢ (𝜑 → 𝐶 ∈ Cat)

Proof of Theorem thinccd

Step	Hyp	Ref	Expression
1		thinccd.c	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
2		thincc 49308	. 2 ⊢ (𝐶 ∈ ThinCat → 𝐶 ∈ Cat)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2108  Catccat 17676  ThinCatcthinc 49303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-nul 5276

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2539  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6484  df-fv 6539  df-ov 7408  df-thinc 49304

This theorem is referenced by:  thincid  49318  thincmon  49319  thincepi  49320  oppcthinco  49325  functhinclem4  49333  functhinc  49334  thincciso  49339  thinccisod  49340  thincsect  49353  thincinv  49355  thinciso  49356  thinccic  49357  termccd  49365  arweutermc  49415