Theorem latasym

Description: A lattice ordering is asymmetric. ( eqss analog.) (Contributed by NM, 8-Oct-2011)

	Ref	Expression
Hypotheses	latref.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	latref.l	⊢ ≤ = ( le ‘ 𝐾 )
Assertion	latasym	⊢ ( ( 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	1 2	latasymb	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) ↔ 𝑋 = 𝑌 ) )
4	3	biimpd	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) → 𝑋 = 𝑌 ) )