Theorem latref

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

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

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	posref	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ≤ 𝑋 )
5	3 4	sylan	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ≤ 𝑋 )