Theorem atlt

Description: Two atoms are unequal iff their join is greater than one of them. (Contributed by NM, 6-May-2012)

	Ref	Expression
Hypotheses	atlt.s	⊢ < = ( lt ‘ 𝐾 )
	atlt.j	⊢ ∨ = ( join ‘ 𝐾 )
	atlt.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	atlt	⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 < ( 𝑃 ∨ 𝑄 ) ↔ 𝑃 ≠ 𝑄 ) )

Proof

Step	Hyp	Ref	Expression
1		atlt.s	⊢ < = ( lt ‘ 𝐾 )
2		atlt.j	⊢ ∨ = ( join ‘ 𝐾 )
3		atlt.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		simp1	⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝐾 ∈ HL )
5		simp2	⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑃 ∈ 𝐴 )
6		simp3	⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑄 ∈ 𝐴 )
7		eqid	⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 )
8	1 2 3 7	atltcvr	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑃 < ( 𝑃 ∨ 𝑄 ) ↔ 𝑃 ( ⋖ ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) )
9	4 5 5 6 8	syl13anc	⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 < ( 𝑃 ∨ 𝑄 ) ↔ 𝑃 ( ⋖ ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) )
10	2 7 3	atcvr1	⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑄 ↔ 𝑃 ( ⋖ ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) )
11	9 10	bitr4d	⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 < ( 𝑃 ∨ 𝑄 ) ↔ 𝑃 ≠ 𝑄 ) )