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Metamath Proof Explorer

Theorem paddasslem1

Description: Lemma for paddass . (Contributed by NM, 8-Jan-2012)

Ref Expression
Hypotheses paddasslem.l ⊢ ≤ = ( le ‘ 𝐾 )
paddasslem.j ⊢ ∨ = ( join ‘ 𝐾 )
paddasslem.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion paddasslem1 ( ( ( 𝐾 ∈ HL ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑥 ≠ 𝑦 ) ∧ ¬ 𝑟 ≤ ( 𝑥 ∨ 𝑦 ) ) → ¬ 𝑥 ≤ ( 𝑟 ∨ 𝑦 ) )

Proof

Step Hyp Ref Expression
1 paddasslem.l ⊢ ≤ = ( le ‘ 𝐾 )
2 paddasslem.j ⊢ ∨ = ( join ‘ 𝐾 )
3 paddasslem.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 1 2 3 hlatexch2 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑥 ≠ 𝑦 ) → ( 𝑥 ≤ ( 𝑟 ∨ 𝑦 ) → 𝑟 ≤ ( 𝑥 ∨ 𝑦 ) ) )
5 4 con3dimp ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑥 ≠ 𝑦 ) ∧ ¬ 𝑟 ≤ ( 𝑥 ∨ 𝑦 ) ) → ¬ 𝑥 ≤ ( 𝑟 ∨ 𝑦 ) )