Theorem lnssplng1

Description: A line defined by two points X and Y , both on a plane H , is entirely contained in H . First part of Theorem 9.25 of Schwabhauser p. 75. (Contributed by Thierry Arnoux, 17-Jun-2026)

	Ref	Expression
Hypotheses	plngval.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	plngval.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	plngval.1	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	plngval.e	⊢ 𝐸 = ( hlG ‘ 𝐺 )
	plngval.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	lnssplng.h	⊢ ( 𝜑 → 𝐻 ∈ ran 𝐸 )
	lnssplng.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐻 )
	lnssplng.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐻 )
	lnssplng.1	⊢ ( 𝜑 → 𝑋 ≠ 𝑌 )
Assertion	lnssplng1	⊢ ( 𝜑 → ( 𝑋 𝐿 𝑌 ) ⊆ 𝐻 )

Proof

Step	Hyp	Ref	Expression
1		plngval.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		plngval.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
3		plngval.1	⊢ 𝐿 = ( LineG ‘ 𝐺 )
4		plngval.e	⊢ 𝐸 = ( hlG ‘ 𝐺 )
5		plngval.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
6		lnssplng.h	⊢ ( 𝜑 → 𝐻 ∈ ran 𝐸 )
7		lnssplng.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐻 )
8		lnssplng.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐻 )
9		lnssplng.1	⊢ ( 𝜑 → 𝑋 ≠ 𝑌 )
10	1 2 3 4 5 6 7 8 9	lnssplng	⊢ ( 𝜑 → ( ( 𝑋 𝐿 𝑌 ) ⊆ 𝐻 ∧ ∃ 𝑠 ∈ ( 𝑃 ∖ ( 𝑋 𝐿 𝑌 ) ) 𝐻 = ( ( 𝑋 𝐿 𝑌 ) 𝐸 𝑠 ) ) )
11	10	simpld	⊢ ( 𝜑 → ( 𝑋 𝐿 𝑌 ) ⊆ 𝐻 )