Theorem tglnpt

Description: Lines are sets of points. (Contributed by Thierry Arnoux, 17-Oct-2019)

	Ref	Expression
Hypotheses	tglng.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	tglng.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	tglng.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	tglnpt.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tglnpt.a	⊢ ( 𝜑 → 𝐴 ∈ ran 𝐿 )
	tglnpt.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
Assertion	tglnpt	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		tglng.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		tglng.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
3		tglng.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		tglnpt.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		tglnpt.a	⊢ ( 𝜑 → 𝐴 ∈ ran 𝐿 )
6		tglnpt.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
7	1 2 3	tglnunirn	⊢ ( 𝐺 ∈ TarskiG → ∪ ran 𝐿 ⊆ 𝑃 )
8	4 7	syl	⊢ ( 𝜑 → ∪ ran 𝐿 ⊆ 𝑃 )
9		elssuni	⊢ ( 𝐴 ∈ ran 𝐿 → 𝐴 ⊆ ∪ ran 𝐿 )
10	5 9	syl	⊢ ( 𝜑 → 𝐴 ⊆ ∪ ran 𝐿 )
11	10 6	sseldd	⊢ ( 𝜑 → 𝑋 ∈ ∪ ran 𝐿 )
12	8 11	sseldd	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )