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 } )