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Theorem exmidomniim 6981
Description: Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 6982. (Contributed by Jim Kingdon, 29-Jun-2022.)
Assertion
Ref Expression
exmidomniim (EXMID → ∀𝑥 𝑥 ∈ Omni)

Proof of Theorem exmidomniim
Dummy variables 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exmidexmid 4090 . . . . . . . . 9 (EXMIDDECID𝑦𝑥 (𝑓𝑦) = 1o)
2 exmiddc 806 . . . . . . . . 9 (DECID𝑦𝑥 (𝑓𝑦) = 1o → (∀𝑦𝑥 (𝑓𝑦) = 1o ∨ ¬ ∀𝑦𝑥 (𝑓𝑦) = 1o))
31, 2syl 14 . . . . . . . 8 (EXMID → (∀𝑦𝑥 (𝑓𝑦) = 1o ∨ ¬ ∀𝑦𝑥 (𝑓𝑦) = 1o))
43orcomd 703 . . . . . . 7 (EXMID → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1o ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o))
54adantr 274 . . . . . 6 ((EXMID𝑓:𝑥⟶2o) → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1o ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o))
6 ffvelrn 5521 . . . . . . . . . . . . . 14 ((𝑓:𝑥⟶2o𝑦𝑥) → (𝑓𝑦) ∈ 2o)
7 df2o3 6295 . . . . . . . . . . . . . 14 2o = {∅, 1o}
86, 7eleqtrdi 2210 . . . . . . . . . . . . 13 ((𝑓:𝑥⟶2o𝑦𝑥) → (𝑓𝑦) ∈ {∅, 1o})
9 elpri 3520 . . . . . . . . . . . . 13 ((𝑓𝑦) ∈ {∅, 1o} → ((𝑓𝑦) = ∅ ∨ (𝑓𝑦) = 1o))
108, 9syl 14 . . . . . . . . . . . 12 ((𝑓:𝑥⟶2o𝑦𝑥) → ((𝑓𝑦) = ∅ ∨ (𝑓𝑦) = 1o))
1110ord 698 . . . . . . . . . . 11 ((𝑓:𝑥⟶2o𝑦𝑥) → (¬ (𝑓𝑦) = ∅ → (𝑓𝑦) = 1o))
1211ralimdva 2476 . . . . . . . . . 10 (𝑓:𝑥⟶2o → (∀𝑦𝑥 ¬ (𝑓𝑦) = ∅ → ∀𝑦𝑥 (𝑓𝑦) = 1o))
1312con3d 605 . . . . . . . . 9 (𝑓:𝑥⟶2o → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1o → ¬ ∀𝑦𝑥 ¬ (𝑓𝑦) = ∅))
1413adantl 275 . . . . . . . 8 ((EXMID𝑓:𝑥⟶2o) → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1o → ¬ ∀𝑦𝑥 ¬ (𝑓𝑦) = ∅))
15 exmidexmid 4090 . . . . . . . . . 10 (EXMIDDECID𝑦𝑥 (𝑓𝑦) = ∅)
16 dfrex2dc 2405 . . . . . . . . . 10 (DECID𝑦𝑥 (𝑓𝑦) = ∅ → (∃𝑦𝑥 (𝑓𝑦) = ∅ ↔ ¬ ∀𝑦𝑥 ¬ (𝑓𝑦) = ∅))
1715, 16syl 14 . . . . . . . . 9 (EXMID → (∃𝑦𝑥 (𝑓𝑦) = ∅ ↔ ¬ ∀𝑦𝑥 ¬ (𝑓𝑦) = ∅))
1817adantr 274 . . . . . . . 8 ((EXMID𝑓:𝑥⟶2o) → (∃𝑦𝑥 (𝑓𝑦) = ∅ ↔ ¬ ∀𝑦𝑥 ¬ (𝑓𝑦) = ∅))
1914, 18sylibrd 168 . . . . . . 7 ((EXMID𝑓:𝑥⟶2o) → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1o → ∃𝑦𝑥 (𝑓𝑦) = ∅))
2019orim1d 761 . . . . . 6 ((EXMID𝑓:𝑥⟶2o) → ((¬ ∀𝑦𝑥 (𝑓𝑦) = 1o ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o) → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o)))
215, 20mpd 13 . . . . 5 ((EXMID𝑓:𝑥⟶2o) → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o))
2221ex 114 . . . 4 (EXMID → (𝑓:𝑥⟶2o → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o)))
2322alrimiv 1830 . . 3 (EXMID → ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o)))
24 isomni 6976 . . . 4 (𝑥 ∈ V → (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o))))
2524elv 2664 . . 3 (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o)))
2623, 25sylibr 133 . 2 (EXMID𝑥 ∈ Omni)
2726alrimiv 1830 1 (EXMID → ∀𝑥 𝑥 ∈ Omni)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 682  DECID wdc 804  wal 1314   = wceq 1316  wcel 1465  wral 2393  wrex 2394  Vcvv 2660  c0 3333  {cpr 3498  EXMIDwem 4088  wf 5089  cfv 5093  1oc1o 6274  2oc2o 6275  Omnicomni 6972
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-exmid 4089  df-id 4185  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-1o 6281  df-2o 6282  df-omni 6974
This theorem is referenced by:  exmidomni  6982
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