Theorem termcbas2

Description: The base of a terminal category is given by its object. (Contributed by Zhi Wang, 20-Oct-2025)

	Ref	Expression
Hypotheses	termcbas.c	|- ( ph -> C e. TermCat )
	termcbas.b	|- B = ( Base ` C )
	termcbasmo.x	|- ( ph -> X e. B )
Assertion	termcbas2	|- ( ph -> B = { X } )

Proof

Step	Hyp	Ref	Expression
1		termcbas.c	|- ( ph -> C e. TermCat )
2		termcbas.b	|- B = ( Base ` C )
3		termcbasmo.x	|- ( ph -> X e. B )
4	1 2	termcbas	|- ( ph -> E. x B = { x } )
5		simpr	|- ( ( ph /\ B = { x } ) -> B = { x } )
6	3	adantr	|- ( ( ph /\ B = { x } ) -> X e. B )
7	6 5	eleqtrd	|- ( ( ph /\ B = { x } ) -> X e. { x } )
8		elsni	|- ( X e. { x } -> X = x )
9	8	sneqd	|- ( X e. { x } -> { X } = { x } )
10	7 9	syl	|- ( ( ph /\ B = { x } ) -> { X } = { x } )
11	5 10	eqtr4d	|- ( ( ph /\ B = { x } ) -> B = { X } )
12	4 11	exlimddv	|- ( ph -> B = { X } )