Theorem omlop 40015

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 40014	. 2 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OL)
2		olop 39988	. 2 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
3	1, 2	syl 18	1 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)

Syntax hints:   → wi 4   ∈ wcel 2143  OPcops 39946  OLcol 39948  OMLcoml 39949

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-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-ol 39952  df-oml 39953

This theorem is referenced by:  omllaw2N  40018  omllaw4  40020  cmtcomlemN  40022  cmt2N  40024  cmt3N  40025  cmt4N  40026  cmtbr2N  40027  cmtbr3N  40028  cmtbr4N  40029  lecmtN  40030  omlfh1N  40032  omlfh3N  40033  omlspjN  40035  atlatmstc  40093