Theorem tgldim0itv

Description: In dimension zero, any two points are equal. (Contributed by Thierry Arnoux, 12-Apr-2019)

	Ref	Expression
Hypotheses	tgbtwndiff.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	tgbtwndiff.d	⊢ − = ( dist ‘ 𝐺 )
	tgbtwndiff.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	tgbtwndiff.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tgbtwndiff.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	tgbtwndiff.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	tgldim0itv.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	tgldim0itv.p	⊢ ( 𝜑 → ( ♯ ‘ 𝑃 ) = 1 )
Assertion	tgldim0itv	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐵 𝐼 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		tgbtwndiff.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		tgbtwndiff.d	⊢ − = ( dist ‘ 𝐺 )
3		tgbtwndiff.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		tgbtwndiff.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		tgbtwndiff.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
6		tgbtwndiff.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
7		tgldim0itv.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
8		tgldim0itv.p	⊢ ( 𝜑 → ( ♯ ‘ 𝑃 ) = 1 )
9	1 8 5 6	tgldim0eq	⊢ ( 𝜑 → 𝐴 = 𝐵 )
10	1 2 3 4 6 7	tgbtwntriv1	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐵 𝐼 𝐶 ) )
11	9 10	eqeltrd	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐵 𝐼 𝐶 ) )