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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 ∧ ( 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 )