Theorem olposN 39510

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 39509	. 2 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
2		opposet 39476	. 2 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset)
3	1, 2	syl 17	1 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Poset)

Syntax hints:   → wi 4   ∈ wcel 2114  Posetcpo 18232  OPcops 39467  OLcol 39469

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 2707  ax-nul 5250

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 2714  df-cleq 2727  df-clel 2810  df-ne 2932  df-ral 3051  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-dm 5633  df-iota 6447  df-fv 6499  df-ov 7361  df-oposet 39471  df-ol 39473

This theorem is referenced by: (None)