Theorem isolat 39212

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 39178	. 2 ⊢ OL = (Lat ∩ OP)
2	1	elin2 4169	1 ⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  Latclat 18397  OPcops 39172  OLcol 39174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-in 3924  df-ol 39178

This theorem is referenced by:  ollat  39213  olop  39214  isolatiN  39216