Theorem isolat 40093

Description: The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)

Assertion

Ref	Expression
isolat	⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))

Proof of Theorem isolat

Step	Hyp	Ref	Expression
1		df-ol 40059	. 2 ⊢ OL = (Lat ∩ OP)
2	1	elin2 4152	1 ⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  Latclat 18525  OPcops 40053  OLcol 40055

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-in 3909  df-ol 40059

This theorem is used by:  ollat  40094  olop  40095  isolatiN  40097