Theorem 2atneat

Description: The join of two distinct atoms is not an atom. (Contributed by NM, 12-Oct-2012)

	Ref	Expression
Hypotheses	2atneat.j	⊢ ∨ = ( join ‘ 𝐾 )
	2atneat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	2atneat	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ¬ ( 𝑃 ∨ 𝑄 ) ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		2atneat.j	⊢ ∨ = ( join ‘ 𝐾 )
2		2atneat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		simpl	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝐾 ∈ HL )
4		simpr1	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑃 ∈ 𝐴 )
5		simpr2	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑄 ∈ 𝐴 )
6		simpr3	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑃 ≠ 𝑄 )
7		eqid	⊢ ( LLines ‘ 𝐾 ) = ( LLines ‘ 𝐾 )
8	1 2 7	llni2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑄 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( LLines ‘ 𝐾 ) )
9	3 4 5 6 8	syl31anc	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ( 𝑃 ∨ 𝑄 ) ∈ ( LLines ‘ 𝐾 ) )
10	2 7	llnneat	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( LLines ‘ 𝐾 ) ) → ¬ ( 𝑃 ∨ 𝑄 ) ∈ 𝐴 )
11	9 10	syldan	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ¬ ( 𝑃 ∨ 𝑄 ) ∈ 𝐴 )