Theorem isolatiN 39314

Description: Properties that determine an ortholattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
isolati.1	⊢ 𝐾 ∈ Lat
isolati.2	⊢ 𝐾 ∈ OP

Assertion

Ref	Expression
isolatiN	⊢ 𝐾 ∈ OL

Proof of Theorem isolatiN

Step	Hyp	Ref	Expression
1		isolati.1	. 2 ⊢ 𝐾 ∈ Lat
2		isolati.2	. 2 ⊢ 𝐾 ∈ OP
3		isolat 39310	. 2 ⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))
4	1, 2, 3	mpbir2an 711	1 ⊢ 𝐾 ∈ OL

Syntax hints:   ∈ wcel 2111  Latclat 18337  OPcops 39270  OLcol 39272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-in 3904  df-ol 39276

This theorem is referenced by: (None)