Theorem olposN 40030

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 40029	. 2 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
2		opposet 39996	. 2 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset)
3	1, 2	syl 18	1 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Poset)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  Posetcpo 18388  OPcops 39987  OLcol 39989

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-nul 5274

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-dm 5676  df-iota 6499  df-fv 6551  df-ov 7426  df-oposet 39991  df-ol 39993

This theorem is used by: (None)