Theorem lpowlpo 7166

Description: LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7165. There is an analogue in terms of analytic omniscience principles at tridceq 14807. (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 7165	. 2 ⊢ (ω ∈ Omni ↔ (ω ∈ WOmni ∧ ω ∈ Markov))
2	1	simplbi 274	1 ⊢ (ω ∈ Omni → ω ∈ WOmni)

Syntax hints:   → wi 4   ∈ wcel 2148  ωcom 4590  Omnicomni 7132  Markovcmarkov 7149  WOmnicwomni 7161

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-nul 4130

This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2740  df-dif 3132  df-un 3134  df-nul 3424  df-sn 3599  df-suc 4372  df-fn 5220  df-f 5221  df-1o 6417  df-omni 7133  df-markov 7150  df-womni 7162

This theorem is referenced by: (None)