Theorem lpowlpo 7291

Description: LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7290. There is an analogue in terms of analytic omniscience principles at tridceq 16167. (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 7290	. 2 ⊢ (ω ∈ Omni ↔ (ω ∈ WOmni ∧ ω ∈ Markov))
2	1	simplbi 274	1 ⊢ (ω ∈ Omni → ω ∈ WOmni)

Syntax hints:   → wi 4   ∈ wcel 2177  ωcom 4651  Omnicomni 7257  Markovcmarkov 7274  WOmnicwomni 7286

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-nul 4181

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-v 2775  df-dif 3172  df-un 3174  df-nul 3465  df-sn 3644  df-suc 4431  df-fn 5288  df-f 5289  df-1o 6520  df-omni 7258  df-markov 7275  df-womni 7287

This theorem is referenced by:  nnnninfen  16130