Theorem 2llnne2N

Description: Condition implying that two intersecting lines are different. (Contributed by NM, 13-Jun-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	2lnne.l	⊢ ≤ = ( le ‘ 𝐾 )
	2lnne.j	⊢ ∨ = ( join ‘ 𝐾 )
	2lnne.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	2llnne2N	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ¬ 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ) → ( 𝑅 ∨ 𝑃 ) ≠ ( 𝑅 ∨ 𝑄 ) )

Proof

Step	Hyp	Ref	Expression
1		2lnne.l	⊢ ≤ = ( le ‘ 𝐾 )
2		2lnne.j	⊢ ∨ = ( join ‘ 𝐾 )
3		2lnne.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		simpl	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝐾 ∈ HL )
5		simprr	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑅 ∈ 𝐴 )
6		simprl	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑃 ∈ 𝐴 )
7	1 2 3	hlatlej2	⊢ ( ( 𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) → 𝑃 ≤ ( 𝑅 ∨ 𝑃 ) )
8	4 5 6 7	syl3anc	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑃 ≤ ( 𝑅 ∨ 𝑃 ) )
9		breq2	⊢ ( ( 𝑅 ∨ 𝑃 ) = ( 𝑅 ∨ 𝑄 ) → ( 𝑃 ≤ ( 𝑅 ∨ 𝑃 ) ↔ 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ) )
10	8 9	syl5ibcom	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ( 𝑅 ∨ 𝑃 ) = ( 𝑅 ∨ 𝑄 ) → 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ) )
11	10	necon3bd	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ¬ 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) → ( 𝑅 ∨ 𝑃 ) ≠ ( 𝑅 ∨ 𝑄 ) ) )
12	11	3impia	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ¬ 𝑃 ≤ ( 𝑅 ∨ 𝑄 ) ) → ( 𝑅 ∨ 𝑃 ) ≠ ( 𝑅 ∨ 𝑄 ) )