Theorem thinccd 48825

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 48824	. 2 ⊢ (𝐶 ∈ ThinCat → 𝐶 ∈ Cat)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∈ wcel 2106  Catccat 17709  ThinCatcthinc 48819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-thinc 48820

This theorem is referenced by:  thincid  48833  thincmon  48834  thincepi  48835  functhinclem4  48844  functhinc  48845  thincciso  48849  thinccisod  48850  thincsect  48858  thincinv  48860  thinciso  48861  thinccic  48862