Theorem lpowlpo 7227

Description: LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7226. There is an analogue in terms of analytic omniscience principles at tridceq 15546. (Contributed by Jim Kingdon, 24-Jul-2024.)

Assertion

Ref	Expression
lpowlpo	⊢ (ω ∈ Omni → ω ∈ WOmni)

Proof of Theorem lpowlpo

Step	Hyp	Ref	Expression
1		omniwomnimkv 7226	. 2 ⊢ (ω ∈ Omni ↔ (ω ∈ WOmni ∧ ω ∈ Markov))
2	1	simplbi 274	1 ⊢ (ω ∈ Omni → ω ∈ WOmni)

Syntax hints:   → wi 4   ∈ wcel 2164  ωcom 4622  Omnicomni 7193  Markovcmarkov 7210  WOmnicwomni 7222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-nul 4155

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-nul 3447  df-sn 3624  df-suc 4402  df-fn 5257  df-f 5258  df-1o 6469  df-omni 7194  df-markov 7211  df-womni 7223

This theorem is referenced by:  nnnninfen  15511