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Metamath Proof Explorer

Theorem latnlej2l

Description: An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012)

Ref Expression
Hypotheses latlej.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
latlej.l ⊢ ≤ = ( le ‘ 𝐾 )
latlej.j ⊢ ∨ = ( join ‘ 𝐾 )
Assertion latnlej2l ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ ¬ 𝑋 ≤ ( 𝑌 ∨ 𝑍 ) ) → ¬ 𝑋 ≤ 𝑌 )

Proof

Step Hyp Ref Expression
1 latlej.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 latlej.l ⊢ ≤ = ( le ‘ 𝐾 )
3 latlej.j ⊢ ∨ = ( join ‘ 𝐾 )
4 1 2 3 latnlej2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ ¬ 𝑋 ≤ ( 𝑌 ∨ 𝑍 ) ) → ( ¬ 𝑋 ≤ 𝑌 ∧ ¬ 𝑋 ≤ 𝑍 ) )
5 4 simpld ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ ¬ 𝑋 ≤ ( 𝑌 ∨ 𝑍 ) ) → ¬ 𝑋 ≤ 𝑌 )