Theorem thinccd 50357

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 50356	. 2 ⊢ (𝐶 ∈ ThinCat → 𝐶 ∈ Cat)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Catccat 17758  ThinCatcthinc 50351

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 2734  ax-nul 5267

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 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-ov 7420  df-thinc 50352

This theorem is used by:  thincid  50366  thincmon  50367  thincepi  50368  oppcthinco  50373  functhinclem4  50381  functhinc  50382  thincciso  50387  thinccisod  50388  thincsect  50401  thincinv  50403  thinciso  50404  thinccic  50405  termccd  50413  arweutermc  50464  funcsn  50475