Theorem olposN 40240

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 40239	. 2 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP)
2		opposet 40206	. 2 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset)
3	1, 2	syl 18	1 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Poset)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Posetcpo 18461  OPcops 40197  OLcol 40199

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 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

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 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-dm 5661  df-iota 6487  df-fv 6539  df-ov 7415  df-oposet 40201  df-ol 40203

This theorem is used by: (None)