Theorem olposN 39554

Description: An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
olposN	⊢ (𝐾 ∈ OL → 𝐾 ∈ Poset)

Proof of Theorem olposN

Step	Hyp	Ref	Expression
1		olop 39553	. 2 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
2		opposet 39520	. 2 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset)
3	1, 2	syl 17	1 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Poset)

Syntax hints:   → wi 4   ∈ wcel 2114  Posetcpo 18235  OPcops 39511  OLcol 39513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5252

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-dm 5635  df-iota 6449  df-fv 6501  df-ov 7364  df-oposet 39515  df-ol 39517

This theorem is referenced by: (None)