Theorem olposN 39714

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 39713	. 2 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
2		opposet 39680	. 2 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset)
3	1, 2	syl 17	1 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Poset)

Syntax hints:   → wi 4   ∈ wcel 2119  Posetcpo 18271  OPcops 39671  OLcol 39673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-dm 5635  df-iota 6448  df-fv 6500  df-ov 7366  df-oposet 39675  df-ol 39677

This theorem is referenced by: (None)