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Theorem omniwomnimkv 7212
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 2767 . 2 (𝐴 ∈ Omni → 𝐴 ∈ V)
2 simpl 109 . . 3 ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) → 𝐴 ∈ WOmni)
32elexd 2769 . 2 ((𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov) → 𝐴 ∈ V)
4 1n0 6472 . . . . . . . . . . . . . . 15 1o ≠ ∅
54nesymi 2406 . . . . . . . . . . . . . 14 ¬ ∅ = 1o
6 eqeq1 2196 . . . . . . . . . . . . . 14 ((𝑓𝑥) = ∅ → ((𝑓𝑥) = 1o ↔ ∅ = 1o))
75, 6mtbiri 676 . . . . . . . . . . . . 13 ((𝑓𝑥) = ∅ → ¬ (𝑓𝑥) = 1o)
87reximi 2587 . . . . . . . . . . . 12 (∃𝑥𝐴 (𝑓𝑥) = ∅ → ∃𝑥𝐴 ¬ (𝑓𝑥) = 1o)
9 rexnalim 2479 . . . . . . . . . . . 12 (∃𝑥𝐴 ¬ (𝑓𝑥) = 1o → ¬ ∀𝑥𝐴 (𝑓𝑥) = 1o)
108, 9syl 14 . . . . . . . . . . 11 (∃𝑥𝐴 (𝑓𝑥) = ∅ → ¬ ∀𝑥𝐴 (𝑓𝑥) = 1o)
1110orim1i 761 . . . . . . . . . 10 ((∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o) → (¬ ∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o))
1211orcomd 730 . . . . . . . . 9 ((∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o) → (∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ¬ ∀𝑥𝐴 (𝑓𝑥) = 1o))
13 df-dc 836 . . . . . . . . 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 730 . . . . . . . 8 (((𝐴 ∈ V ∧ 𝑓:𝐴⟶2o) ∧ (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o)) → (∀𝑥𝐴 (𝑓𝑥) = 1o ∨ ∃𝑥𝐴 (𝑓𝑥) = ∅))
1817ord 725 . . . . . . 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 789 . . . . . . . 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 730 . . . . . 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 1835 . . 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 7181 . . 3 (𝐴 ∈ V → (𝐴 ∈ Omni ↔ ∀𝑓(𝑓:𝐴⟶2o → (∃𝑥𝐴 (𝑓𝑥) = ∅ ∨ ∀𝑥𝐴 (𝑓𝑥) = 1o))))
30 iswomni 7210 . . . . . 6 (𝐴 ∈ V → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓:𝐴⟶2oDECID𝑥𝐴 (𝑓𝑥) = 1o)))
31 ismkv 7198 . . . . . 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 1492 . . . . 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 1481 . . . 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 705 1 (𝐴 ∈ Omni ↔ (𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709  DECID wdc 835  wal 1362   = wceq 1364  wcel 2160  wral 2468  wrex 2469  Vcvv 2756  c0 3442  wf 5238  cfv 5242  1oc1o 6449  2oc2o 6450  Omnicomni 7179  Markovcmarkov 7196  WOmnicwomni 7208
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 4151
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 2758  df-dif 3151  df-un 3153  df-nul 3443  df-sn 3620  df-suc 4396  df-fn 5245  df-f 5246  df-1o 6456  df-omni 7180  df-markov 7197  df-womni 7209
This theorem is referenced by:  lpowlpo  7213
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