This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem hlrelat1

Description: An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of MaedaMaeda p. 30. ( chpssati , with /\ swapped, analog.) (Contributed by NM, 4-Dec-2011)

Ref Expression
Hypotheses hlrelat1.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
hlrelat1.l ⊢ ≤ = ( le ‘ 𝐾 )
hlrelat1.s ⊢ < = ( lt ‘ 𝐾 )
hlrelat1.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion hlrelat1 ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 hlrelat1.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 hlrelat1.l ⊢ ≤ = ( le ‘ 𝐾 )
3 hlrelat1.s ⊢ < = ( lt ‘ 𝐾 )
4 hlrelat1.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 hlomcmat ⊢ ( 𝐾 ∈ HL → ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) )
6 1 2 3 4 atlrelat1 ⊢ ( ( ( 𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) )
7 5 6 syl3an1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌 ) ) )