Theorem 0thincg

Description: Any structure with an empty set of objects is a thin category. (Contributed by Zhi Wang, 17-Sep-2024)

	Ref	Expression
Assertion	0thincg	⊢ ( ( 𝐶 ∈ 𝑉 ∧ ∅ = ( Base ‘ 𝐶 ) ) → 𝐶 ∈ ThinCat )

Proof

Step	Hyp	Ref	Expression
1		0catg	⊢ ( ( 𝐶 ∈ 𝑉 ∧ ∅ = ( Base ‘ 𝐶 ) ) → 𝐶 ∈ Cat )
2		ral0	⊢ ∀ 𝑥 ∈ ∅ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∃* 𝑓 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 )
3		raleq	⊢ ( ∅ = ( Base ‘ 𝐶 ) → ( ∀ 𝑥 ∈ ∅ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∃* 𝑓 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∃* 𝑓 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) )
4	2 3	mpbii	⊢ ( ∅ = ( Base ‘ 𝐶 ) → ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∃* 𝑓 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) )
5	4	adantl	⊢ ( ( 𝐶 ∈ 𝑉 ∧ ∅ = ( Base ‘ 𝐶 ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∃* 𝑓 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) )
6		eqid	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
7		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
8	6 7	isthinc	⊢ ( 𝐶 ∈ ThinCat ↔ ( 𝐶 ∈ Cat ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∃* 𝑓 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) )
9	1 5 8	sylanbrc	⊢ ( ( 𝐶 ∈ 𝑉 ∧ ∅ = ( Base ‘ 𝐶 ) ) → 𝐶 ∈ ThinCat )