Theorem mndtcbaseu

Description: The category built from a monoid contains precisely one object. (Contributed by Zhi Wang, 22-Sep-2024)

	Ref	Expression
Hypotheses	mndtcbaseu.c	|- ( ph -> C = ( MndToCat ` M ) )
	mndtcbaseu.m	|- ( ph -> M e. Mnd )
	mndtcbaseu.b	|- ( ph -> B = ( Base ` C ) )
Assertion	mndtcbaseu	|- ( ph -> E! x x e. B )

Proof

Step	Hyp	Ref	Expression
1		mndtcbaseu.c	|- ( ph -> C = ( MndToCat ` M ) )
2		mndtcbaseu.m	|- ( ph -> M e. Mnd )
3		mndtcbaseu.b	|- ( ph -> B = ( Base ` C ) )
4	1 2 3	mndtcbasval	|- ( ph -> B = { M } )
5		sneq	|- ( x = M -> { x } = { M } )
6	5	eqeq2d	|- ( x = M -> ( B = { x } <-> B = { M } ) )
7	2 4 6	spcedv	|- ( ph -> E. x B = { x } )
8		eusn	|- ( E! x x e. B <-> E. x B = { x } )
9	7 8	sylibr	|- ( ph -> E! x x e. B )