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 )