Theorem atnlt

Description: Two atoms cannot satisfy the less than relation. (Contributed by NM, 7-Feb-2012)

	Ref	Expression
Hypotheses	atnlt.s	⊢ < = ( lt ‘ 𝐾 )
	atnlt.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	atnlt	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ¬ 𝑃 < 𝑄 )

Proof

Step	Hyp	Ref	Expression
1		atnlt.s	⊢ < = ( lt ‘ 𝐾 )
2		atnlt.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3	1	pltirr	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ) → ¬ 𝑃 < 𝑃 )
4	3	3adant3	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ¬ 𝑃 < 𝑃 )
5		breq2	⊢ ( 𝑃 = 𝑄 → ( 𝑃 < 𝑃 ↔ 𝑃 < 𝑄 ) )
6	5	notbid	⊢ ( 𝑃 = 𝑄 → ( ¬ 𝑃 < 𝑃 ↔ ¬ 𝑃 < 𝑄 ) )
7	4 6	syl5ibcom	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 = 𝑄 → ¬ 𝑃 < 𝑄 ) )
8		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
9	8 1	pltle	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 < 𝑄 → 𝑃 ( le ‘ 𝐾 ) 𝑄 ) )
10	8 2	atcmp	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ( le ‘ 𝐾 ) 𝑄 ↔ 𝑃 = 𝑄 ) )
11	9 10	sylibd	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 < 𝑄 → 𝑃 = 𝑄 ) )
12	11	necon3ad	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑄 → ¬ 𝑃 < 𝑄 ) )
13	7 12	pm2.61dne	⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ¬ 𝑃 < 𝑄 )