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Metamath Proof Explorer

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 )