Theorem omlop 40075

Description: An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)

Assertion

Ref	Expression
omlop	⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)

Proof of Theorem omlop

Step	Hyp	Ref	Expression
1		omlol 40074	. 2 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)
2		olop 40048	. 2 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
3	1, 2	syl 18	1 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  OPcops 40006  OLcol 40008  OMLcoml 40009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7422  df-ol 40012  df-oml 40013

This theorem is used by:  omllaw2N  40078  omllaw4  40080  cmtcomlemN  40082  cmt2N  40084  cmt3N  40085  cmt4N  40086  cmtbr2N  40087  cmtbr3N  40088  cmtbr4N  40089  lecmtN  40090  omlfh1N  40092  omlfh3N  40093  omlspjN  40095  atlatmstc  40153