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 )