termchomn0

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

	Ref	Expression
Hypotheses	termcbas.c	No typesetting found for \|- ( ph -> C e. TermCat ) with typecode \|-
	termcbas.b	$⊢ B = {Base}_{C}$
	termcbasmo.x	$⊢ φ \to X \in B$
	termcbasmo.y	$⊢ φ \to Y \in B$
	termcid.h	$⊢ H = Hom (C)$
Assertion	termchomn0	$⊢ φ \to \neg X H Y = \emptyset$

Step	Hyp	Ref	Expression
1		termcbas.c	Could not format ( ph -> C e. TermCat ) : No typesetting found for \|- ( ph -> C e. TermCat ) with typecode \|-
2		termcbas.b	$⊢ B = {Base}_{C}$
3		termcbasmo.x	$⊢ φ \to X \in B$
4		termcbasmo.y	$⊢ φ \to Y \in B$
5		termcid.h	$⊢ H = Hom (C)$
6		eqid	$⊢ Id (C) = Id (C)$
7	1	termccd	$⊢ φ \to C \in Cat$
8	2 5 6 7 3	catidcl	$⊢ φ \to Id (C) (X) \in (X H X)$
9	1 2 3 4	termcbasmo	$⊢ φ \to X = Y$
10	9	oveq2d	$⊢ φ \to X H X = X H Y$
11	8 10	eleqtrd	$⊢ φ \to Id (C) (X) \in (X H Y)$
12		n0i	$⊢ Id (C) (X) \in (X H Y) \to \neg X H Y = \emptyset$
13	11 12	syl	$⊢ φ \to \neg X H Y = \emptyset$

Metamath Proof Explorer