Theorem isolat 34499

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 34465	. 2 ⊢ OL = (Lat ∩ OP)
2	1	elin2 3801	1 ⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))

Syntax hints:   ↔ wb 196   ∧ wa 384   ∈ wcel 1990  Latclat 17045  OPcops 34459  OLcol 34461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ol 34465

This theorem is referenced by:  ollat  34500  olop  34501  isolatiN  34503