Theorem lpowlpo 7132

Description: LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7131. There is an analogue in terms of analytic omniscience principles at tridceq 13935. (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 7131	. 2 ⊢ (ω ∈ Omni ↔ (ω ∈ WOmni ∧ ω ∈ Markov))
2	1	simplbi 272	1 ⊢ (ω ∈ Omni → ω ∈ WOmni)

Syntax hints:   → wi 4   ∈ wcel 2136  ωcom 4567  Omnicomni 7098  Markovcmarkov 7115  WOmnicwomni 7127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-nul 4108

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-nul 3410  df-sn 3582  df-suc 4349  df-fn 5191  df-f 5192  df-1o 6384  df-omni 7099  df-markov 7116  df-womni 7128

This theorem is referenced by: (None)