Metamath Proof Explorer


Theorem 0subcat

Description: For any category C , the empty set is a (full) subcategory of C , see example 4.3(1.a) in Adamek p. 48. (Contributed by AV, 23-Apr-2020)

Ref Expression
Assertion 0subcat C Cat Subcat C

Proof

Step Hyp Ref Expression
1 0ssc C Cat cat Hom 𝑓 C
2 ral0 x Id C x x x y z f x y g y z g x y comp C z f x z
3 2 a1i C Cat x Id C x x x y z f x y g y z g x y comp C z f x z
4 eqid Hom 𝑓 C = Hom 𝑓 C
5 eqid Id C = Id C
6 eqid comp C = comp C
7 id C Cat C Cat
8 f0 :
9 ffn : Fn
10 8 9 ax-mp Fn
11 0xp × =
12 11 fneq2i Fn × Fn
13 10 12 mpbir Fn ×
14 13 a1i C Cat Fn ×
15 4 5 6 7 14 issubc2 C Cat Subcat C cat Hom 𝑓 C x Id C x x x y z f x y g y z g x y comp C z f x z
16 1 3 15 mpbir2and C Cat Subcat C