Theorem isolat 39582

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 39548	. 2 ⊢ OL = (Lat ∩ OP)
2	1	elin2 4157	1 ⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  Latclat 18366  OPcops 39542  OLcol 39544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-in 3910  df-ol 39548

This theorem is referenced by:  ollat  39583  olop  39584  isolatiN  39586