Theorem termchomn0

Description: All hom-sets of a terminal category are non-empty. (Contributed by Zhi Wang, 17-Oct-2025)

	Ref	Expression
Hypotheses	termcbas.c	|- ( ph -> C e. TermCat )
	termcbas.b	|- B = ( Base ` C )
	termcbasmo.x	|- ( ph -> X e. B )
	termcbasmo.y	|- ( ph -> Y e. B )
	termcid.h	|- H = ( Hom ` C )
Assertion	termchomn0	|- ( ph -> -. ( X H Y ) = (/) )

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		termcbasmo.y	|- ( ph -> Y e. B )
5		termcid.h	|- H = ( Hom ` C )
6		eqid	|- ( Id ` C ) = ( Id ` C )
7	1	termccd	|- ( ph -> C e. Cat )
8	2 5 6 7 3	catidcl	|- ( ph -> ( ( Id ` C ) ` X ) e. ( X H X ) )
9	1 2 3 4	termcbasmo	|- ( ph -> X = Y )
10	9	oveq2d	|- ( ph -> ( X H X ) = ( X H Y ) )
11	8 10	eleqtrd	|- ( ph -> ( ( Id ` C ) ` X ) e. ( X H Y ) )
12		n0i	|- ( ( ( Id ` C ) ` X ) e. ( X H Y ) -> -. ( X H Y ) = (/) )
13	11 12	syl	|- ( ph -> -. ( X H Y ) = (/) )