Theorem isolatiN 35372

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 35368	. 2 ⊢ (𝐾 ∈ OL ↔ (𝐾 ∈ Lat ∧ 𝐾 ∈ OP))
4	1, 2, 3	mpbir2an 701	1 ⊢ 𝐾 ∈ OL

Syntax hints:   ∈ wcel 2107  Latclat 17431  OPcops 35328  OLcol 35330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-v 3400  df-in 3799  df-ol 35334

This theorem is referenced by: (None)