Theorem isolatiN 39968

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 39964	. 2 ⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))
4	1, 2, 3	mpbir2an 723	1 ⊢ 𝐾 ∈ OL

Syntax hints:   ∈ wcel 2143  Latclat 18488  OPcops 39924  OLcol 39926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-in 3913  df-ol 39930

This theorem is referenced by: (None)