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Theorem omniwomnimkv 7159
Description: A set is omniscient if and only if it is weakly omniscient and Markov. The case 𝐴 = ω says that LPO WLPO MP which is a remark following Definition 2.5 of [Pierik], p. 9. (Contributed by Jim Kingdon, 9-Jun-2024.)
Assertion
Ref Expression
omniwomnimkv (𝐴 ∈ Omni ↔ (𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov))

Proof of Theorem omniwomnimkv
Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2748 . 2 (𝐴 ∈ Omni → 𝐴 ∈ V)
2 simpl 109 . . 3 ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) → 𝐴 ∈ WOmni)
32elexd 2750 . 2 ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) → 𝐴 ∈ V)
4 1n0 6427 . . . . . . . . . . . . . . 15 1o ≠ ∅
54nesymi 2393 . . . . . . . . . . . . . 14 ¬ ∅ = 1o
6 eqeq1 2184 . . . . . . . . . . . . . 14 ((𝑓𝑥) = ∅ → ((𝑓𝑥) = 1o ↔ ∅ = 1o))
75, 6mtbiri 675 . . . . . . . . . . . . 13 ((𝑓𝑥) = ∅ → ¬ (𝑓𝑥) = 1o)
87reximi 2574 . . . . . . . . . . . 12 (∃𝑥𝐴 (𝑓𝑥) = ∅ → ∃𝑥𝐴 ¬ (𝑓𝑥) = 1o)
9 rexnalim 2466 . . . . . . . . . . . 12 (∃𝑥𝐴 ¬ (𝑓𝑥) = 1o → ¬ ∀𝑥𝐴 (𝑓𝑥) = 1o)
108, 9syl 14 . . . . . . . . . . 11 (∃𝑥𝐴 (𝑓𝑥) = ∅ → ¬ ∀𝑥𝐴 (𝑓𝑥) = 1o)
1110orim1i 760 . . . . . . . . . 10 ((∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o) → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o))
1211orcomd 729 . . . . . . . . 9 ((∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o) → (∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ¬ ∀𝑥𝐴 (𝑓𝑥) = 1o))
13 df-dc 835 . . . . . . . . 9 (DECID𝑥𝐴 (𝑓𝑥) = 1o ↔ (∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ¬ ∀𝑥𝐴 (𝑓𝑥) = 1o))
1412, 13sylibr 134 . . . . . . . 8 ((∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o) → DECID𝑥𝐴 (𝑓𝑥) = 1o)
1514adantl 277 . . . . . . 7 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)) → DECID𝑥𝐴 (𝑓𝑥) = 1o)
16 simpr 110 . . . . . . . . 9 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)) → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o))
1716orcomd 729 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)) → (∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ∃𝑥𝐴 (𝑓𝑥) = ∅))
1817ord 724 . . . . . . 7 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)) → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))
1915, 18jca 306 . . . . . 6 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)) → (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅)))
20 simprl 529 . . . . . . . . 9 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))) → DECID𝑥𝐴 (𝑓𝑥) = 1o)
2120, 13sylib 122 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))) → (∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ¬ ∀𝑥𝐴 (𝑓𝑥) = 1o))
22 simprr 531 . . . . . . . . 9 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))) → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))
2322orim2d 788 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))) → ((∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ¬ ∀𝑥𝐴 (𝑓𝑥) = 1o) → (∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ∃𝑥𝐴 (𝑓𝑥) = ∅)))
2421, 23mpd 13 . . . . . . 7 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))) → (∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ∃𝑥𝐴 (𝑓𝑥) = ∅))
2524orcomd 729 . . . . . 6 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))) → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o))
2619, 25impbida 596 . . . . 5 ((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) → ((∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o) ↔ (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))))
2726pm5.74da 443 . . . 4 (𝐴 ∈ V → ((𝑓:𝐴⟶2o → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)) ↔ (𝑓:𝐴⟶2o → (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅)))))
2827albidv 1824 . . 3 (𝐴 ∈ V → (∀𝑓(𝑓:𝐴⟶2o → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)) ↔ ∀𝑓(𝑓:𝐴⟶2o → (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅)))))
29 isomni 7128 . . 3 (𝐴 ∈ V → (𝐴 ∈ Omni ↔ ∀𝑓(𝑓:𝐴⟶2o → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o))))
30 iswomni 7157 . . . . . 6 (𝐴 ∈ V → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓:𝐴⟶2oDECID𝑥𝐴 (𝑓𝑥) = 1o)))
31 ismkv 7145 . . . . . 6 (𝐴 ∈ V → (𝐴 ∈ Markov ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))))
3230, 31anbi12d 473 . . . . 5 (𝐴 ∈ V → ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) ↔ (∀𝑓(𝑓:𝐴⟶2oDECID𝑥𝐴 (𝑓𝑥) = 1o) ∧ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅)))))
33 19.26 1481 . . . . 5 (∀𝑓((𝑓:𝐴⟶2oDECID𝑥𝐴 (𝑓𝑥) = 1o) ∧ (𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))) ↔ (∀𝑓(𝑓:𝐴⟶2oDECID𝑥𝐴 (𝑓𝑥) = 1o) ∧ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))))
3432, 33bitr4di 198 . . . 4 (𝐴 ∈ V → ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) ↔ ∀𝑓((𝑓:𝐴⟶2oDECID𝑥𝐴 (𝑓𝑥) = 1o) ∧ (𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅)))))
35 jcab 603 . . . . 5 ((𝑓:𝐴⟶2o → (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))) ↔ ((𝑓:𝐴⟶2oDECID𝑥𝐴 (𝑓𝑥) = 1o) ∧ (𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))))
3635albii 1470 . . . 4 (∀𝑓(𝑓:𝐴⟶2o → (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))) ↔ ∀𝑓((𝑓:𝐴⟶2oDECID𝑥𝐴 (𝑓𝑥) = 1o) ∧ (𝑓:𝐴⟶2o → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅))))
3734, 36bitr4di 198 . . 3 (𝐴 ∈ V → ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) ↔ ∀𝑓(𝑓:𝐴⟶2o → (DECID𝑥𝐴 (𝑓𝑥) = 1o ∧ (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o → ∃𝑥𝐴 (𝑓𝑥) = ∅)))))
3828, 29, 373bitr4d 220 . 2 (𝐴 ∈ V → (𝐴 ∈ Omni ↔ (𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov)))
391, 3, 38pm5.21nii 704 1 (𝐴 ∈ Omni ↔ (𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834  wal 1351   = wceq 1353  wcel 2148  wral 2455  wrex 2456  Vcvv 2737  c0 3422  wf 5208  cfv 5212  1oc1o 6404  2oc2o 6405  Omnicomni 7126  Markovcmarkov 7143  WOmnicwomni 7155
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 4126
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 2739  df-dif 3131  df-un 3133  df-nul 3423  df-sn 3597  df-suc 4368  df-fn 5215  df-f 5216  df-1o 6411  df-omni 7127  df-markov 7144  df-womni 7156
This theorem is referenced by:  lpowlpo  7160
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