Theorem atleneN

Description: Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		atlene.l	⊢ ≤ = ( le ‘ 𝐾 )
2		atlene.j	⊢ ∨ = ( join ‘ 𝐾 )
3		atlene.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		eqid	⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 )
5	1 2 4 3	atcvrj1	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑅 ∧ 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) ) ) → 𝑃 ( ⋖ ‘ 𝐾 ) ( 𝑄 ∨ 𝑅 ) )
6	2 4 3	atcvrneN	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ( ⋖ ‘ 𝐾 ) ( 𝑄 ∨ 𝑅 ) ) → 𝑄 ≠ 𝑅 )
7	5 6	syld3an3	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑅 ∧ 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) ) ) → 𝑄 ≠ 𝑅 )