Theorem omlop 39900

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 39899	. 2 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)
2		olop 39873	. 2 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
3	1, 2	syl 18	1 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)

Syntax hints:   → wi 4   ∈ wcel 2149  OPcops 39831  OLcol 39833  OMLcoml 39834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-iota 6490  df-fv 6542  df-ov 7411  df-ol 39837  df-oml 39838

This theorem is referenced by:  omllaw2N  39903  omllaw4  39905  cmtcomlemN  39907  cmt2N  39909  cmt3N  39910  cmt4N  39911  cmtbr2N  39912  cmtbr3N  39913  cmtbr4N  39914  lecmtN  39915  omlfh1N  39917  omlfh3N  39918  omlspjN  39920  atlatmstc  39978