oppne1

Description: Points lying on opposite sides of a line cannot be on the line. (Contributed by Thierry Arnoux, 3-Mar-2020)

	Ref	Expression
Hypotheses	hpg.p	$⊢ P = {Base}_{G}$
	hpg.d	$⊢ \underset{˙}{-} = dist (G)$
	hpg.i	$⊢ I = Itv (G)$
	hpg.o	$⊢ O = \{⟨a, b⟩ \| ((a \in (P ∖ D) \land b \in (P ∖ D)) \land \exists t \in D t \in (a I b))\}$
	opphl.l	$⊢ L = {Line}_{𝒢} (G)$
	opphl.d	$⊢ φ \to D \in ran L$
	opphl.g	$⊢ φ \to G \in 𝒢_{Tarski}$
	oppcom.a	$⊢ φ \to A \in P$
	oppcom.b	$⊢ φ \to B \in P$
	oppcom.o	$⊢ φ \to A O B$
Assertion	oppne1	$⊢ φ \to \neg A \in D$

Step	Hyp	Ref	Expression
1		hpg.p	$⊢ P = {Base}_{G}$
2		hpg.d	$⊢ \underset{˙}{-} = dist (G)$
3		hpg.i	$⊢ I = Itv (G)$
4		hpg.o	$⊢ O = \{⟨a, b⟩ \| ((a \in (P ∖ D) \land b \in (P ∖ D)) \land \exists t \in D t \in (a I b))\}$
5		opphl.l	$⊢ L = {Line}_{𝒢} (G)$
6		opphl.d	$⊢ φ \to D \in ran L$
7		opphl.g	$⊢ φ \to G \in 𝒢_{Tarski}$
8		oppcom.a	$⊢ φ \to A \in P$
9		oppcom.b	$⊢ φ \to B \in P$
10		oppcom.o	$⊢ φ \to A O B$
11	1 2 3 4 8 9	islnopp	$⊢ φ \to (A O B \leftrightarrow ((\neg A \in D \land \neg B \in D) \land \exists t \in D t \in (A I B)))$
12	10 11	mpbid	$⊢ φ \to ((\neg A \in D \land \neg B \in D) \land \exists t \in D t \in (A I B))$
13	12	simplld	$⊢ φ \to \neg A \in D$

Metamath Proof Explorer