Theorem thinccd 49910

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 49909	. 2 ⊢ (𝐶 ∈ ThinCat → 𝐶 ∈ Cat)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2114  Catccat 17621  ThinCatcthinc 49904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6448  df-fv 6500  df-ov 7363  df-thinc 49905

This theorem is referenced by:  thincid  49919  thincmon  49920  thincepi  49921  oppcthinco  49926  functhinclem4  49934  functhinc  49935  thincciso  49940  thinccisod  49941  thincsect  49954  thincinv  49956  thinciso  49957  thinccic  49958  termccd  49966  arweutermc  50017  funcsn  50028