Theorem thinccatd 50530

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

Hypothesis

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

Assertion

Ref	Expression
thinccatd	⊢ (𝜑 → 𝐶 ∈ Cat)

Proof of Theorem thinccatd

Step	Hyp	Ref	Expression
1		thinccatd.c	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
2		thinccat 50529	. 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 17838  ThinCatcthinc 50524

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

This theorem is used by:  thincid  50539  thincmon  50540  thincepi  50541  oppcthinco  50546  functhinclem4  50554  functhinc  50555  thincciso  50560  thinccisod  50561  thincsect  50574  thincinv  50576  thinciso  50577  thinccic  50578  termccatd  50586  arweutermc  50637  funcsn  50648