Metamath Proof Explorer


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 ∧ ( 𝑃𝐴𝑄𝐴𝑃𝑄 ) ) → ¬ ( 𝑃 𝑄 ) ∈ 𝐴 )

Proof

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