Theorem isolat 39176

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 39142	. 2 ⊢ OL = (Lat ∩ OP)
2	1	elin2 4178	1 ⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  Latclat 18439  OPcops 39136  OLcol 39138

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-in 3933  df-ol 39142

This theorem is referenced by:  ollat  39177  olop  39178  isolatiN  39180