Theorem isolatiN 39555

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 39551	. 2 ⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))
4	1, 2, 3	mpbir2an 712	1 ⊢ 𝐾 ∈ OL

Syntax hints:   ∈ wcel 2114  Latclat 18359  OPcops 39511  OLcol 39513

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 3443  df-in 3909  df-ol 39517

This theorem is referenced by: (None)