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 ) )