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Metamath Proof Explorer

Theorem latnlej

Description: An idiom to express that a lattice element differs from two others. (Contributed by NM, 28-May-2012)

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

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 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → 𝑌 ≤ ( 𝑌 ∨ 𝑍 ) )
5 4 3adant3r1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ≤ ( 𝑌 ∨ 𝑍 ) )
6 breq1 ⊢ ( 𝑋 = 𝑌 → ( 𝑋 ≤ ( 𝑌 ∨ 𝑍 ) ↔ 𝑌 ≤ ( 𝑌 ∨ 𝑍 ) ) )
7 5 6 syl5ibrcom ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 = 𝑌 → 𝑋 ≤ ( 𝑌 ∨ 𝑍 ) ) )
8 7 necon3bd ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ¬ 𝑋 ≤ ( 𝑌 ∨ 𝑍 ) → 𝑋 ≠ 𝑌 ) )
9 1 2 3 latlej2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → 𝑍 ≤ ( 𝑌 ∨ 𝑍 ) )
10 9 3adant3r1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑍 ≤ ( 𝑌 ∨ 𝑍 ) )
11 breq1 ⊢ ( 𝑋 = 𝑍 → ( 𝑋 ≤ ( 𝑌 ∨ 𝑍 ) ↔ 𝑍 ≤ ( 𝑌 ∨ 𝑍 ) ) )
12 10 11 syl5ibrcom ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 = 𝑍 → 𝑋 ≤ ( 𝑌 ∨ 𝑍 ) ) )
13 12 necon3bd ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ¬ 𝑋 ≤ ( 𝑌 ∨ 𝑍 ) → 𝑋 ≠ 𝑍 ) )
14 8 13 jcad ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ¬ 𝑋 ≤ ( 𝑌 ∨ 𝑍 ) → ( 𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ) ) )
15 14 3impia ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ ¬ 𝑋 ≤ ( 𝑌 ∨ 𝑍 ) ) → ( 𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ) )