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Metamath Proof Explorer

Theorem termcbas

Description: The base of a terminal category is a singleton. (Contributed by Zhi Wang, 16-Oct-2025)

Ref Expression
Hypotheses termcbas.c ⊢ ( 𝜑 → 𝐶 ∈ TermCat )
termcbas.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
Assertion termcbas ( 𝜑 → ∃ 𝑥 𝐵 = { 𝑥 } )

Proof

Step Hyp Ref Expression
1 termcbas.c ⊢ ( 𝜑 → 𝐶 ∈ TermCat )
2 termcbas.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
3 2 istermc ⊢ ( 𝐶 ∈ TermCat ↔ ( 𝐶 ∈ ThinCat ∧ ∃ 𝑥 𝐵 = { 𝑥 } ) )
4 1 3 sylib ⊢ ( 𝜑 → ( 𝐶 ∈ ThinCat ∧ ∃ 𝑥 𝐵 = { 𝑥 } ) )
5 4 simprd ⊢ ( 𝜑 → ∃ 𝑥 𝐵 = { 𝑥 } )