Theorem lpowlpo 7184

Description: LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7183. There is an analogue in terms of analytic omniscience principles at tridceq 15202. (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 7183	. 2 ⊢ (ω ∈ Omni ↔ (ω ∈ WOmni ∧ ω ∈ Markov))
2	1	simplbi 274	1 ⊢ (ω ∈ Omni → ω ∈ WOmni)

Syntax hints:   → wi 4   ∈ wcel 2160  ωcom 4604  Omnicomni 7150  Markovcmarkov 7167  WOmnicwomni 7179

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 2171  ax-nul 4144

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 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-nul 3438  df-sn 3613  df-suc 4386  df-fn 5234  df-f 5235  df-1o 6435  df-omni 7151  df-markov 7168  df-womni 7180

This theorem is referenced by: (None)