Theorem latasymb

Description: A lattice ordering is asymmetric. ( eqss analog.) (Contributed by NM, 22-Oct-2011)

	Ref	Expression
Hypotheses	latref.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	latref.l	⊢ ≤ = ( le ‘ 𝐾 )
Assertion	latasymb	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) ↔ 𝑋 = 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		latref.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		latref.l	⊢ ≤ = ( le ‘ 𝐾 )
3		latpos	⊢ ( 𝐾 ∈ Lat → 𝐾 ∈ Poset )
4	1 2	posasymb	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) ↔ 𝑋 = 𝑌 ) )
5	3 4	syl3an1	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) ↔ 𝑋 = 𝑌 ) )