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Metamath Proof Explorer

Theorem llnbase

Description: A lattice line is a lattice element. (Contributed by NM, 16-Jun-2012)

Ref Expression
Hypotheses llnbase.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
llnbase.n ⊢ 𝑁 = ( LLines ‘ 𝐾 )
Assertion llnbase ( 𝑋 ∈ 𝑁 → 𝑋 ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 llnbase.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 llnbase.n ⊢ 𝑁 = ( LLines ‘ 𝐾 )
3 n0i ⊢ ( 𝑋 ∈ 𝑁 → ¬ 𝑁 = ∅ )
4 2 eqeq1i ⊢ ( 𝑁 = ∅ ↔ ( LLines ‘ 𝐾 ) = ∅ )
5 3 4 sylnib ⊢ ( 𝑋 ∈ 𝑁 → ¬ ( LLines ‘ 𝐾 ) = ∅ )
6 fvprc ⊢ ( ¬ 𝐾 ∈ V → ( LLines ‘ 𝐾 ) = ∅ )
7 5 6 nsyl2 ⊢ ( 𝑋 ∈ 𝑁 → 𝐾 ∈ V )
8 eqid ⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 )
9 eqid ⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
10 1 8 9 2 islln ⊢ ( 𝐾 ∈ V → ( 𝑋 ∈ 𝑁 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ ( Atoms ‘ 𝐾 ) 𝑝 ( ⋖ ‘ 𝐾 ) 𝑋 ) ) )
11 10 simprbda ⊢ ( ( 𝐾 ∈ V ∧ 𝑋 ∈ 𝑁 ) → 𝑋 ∈ 𝐵 )
12 7 11 mpancom ⊢ ( 𝑋 ∈ 𝑁 → 𝑋 ∈ 𝐵 )