Theorem thinccd 45922

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 45921	. 2 ⊢ (𝐶 ∈ ThinCat → 𝐶 ∈ Cat)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2112  Catccat 17121  ThinCatcthinc 45916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-nul 5184

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-iota 6316  df-fv 6366  df-ov 7194  df-thinc 45917

This theorem is referenced by:  thincid  45930  thincmon  45931  thincepi  45932  functhinclem4  45941  functhinc  45942  thincciso  45946  thincsect  45954  thincinv  45956  thinciso  45957  thinccic  45958