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	$$⊢\phi \to C=\mathrm{MndToCat}\left(M\right)$$
	mndtcbaseu.m	$$⊢\phi \to M\in \mathrm{Mnd}$$
	mndtcbaseu.b	$$⊢\phi \to B={\mathrm{Base}}_C$$
Assertion	mndtcbaseu	$$⊢\phi \to \exists !xx\in B$$

Proof

Step	Hyp	Ref	Expression
1		mndtcbaseu.c	$$⊢\phi \to C=\mathrm{MndToCat}\left(M\right)$$
2		mndtcbaseu.m	$$⊢\phi \to M\in \mathrm{Mnd}$$
3		mndtcbaseu.b	$$⊢\phi \to B={\mathrm{Base}}_C$$
4	1 2 3	mndtcbasval	$$⊢\phi \to B=\left\{M\right\}$$
5		sneq	$$⊢x=M\to \left\{x\right\}=\left\{M\right\}$$
6	5	eqeq2d	$$⊢x=M\to \left(B=\left\{x\right\}\leftrightarrow B=\left\{M\right\}\right)$$
7	2 4 6	spcedv	$$⊢\phi \to \exists xB=\left\{x\right\}$$
8		eusn	$$⊢\exists !xx\in B\leftrightarrow \exists xB=\left\{x\right\}$$
9	7 8	sylibr	$$⊢\phi \to \exists !xx\in B$$