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Theorem exmidomniim 7168
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 7169. (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 4214 . . . . . . . . 9 (EXMIDDECID𝑦𝑥 (𝑓𝑦) = 1o)
2 exmiddc 837 . . . . . . . . 9 (DECID𝑦𝑥 (𝑓𝑦) = 1o → (∀𝑦𝑥 (𝑓𝑦) = 1o ∨ ¬ ∀𝑦𝑥 (𝑓𝑦) = 1o))
31, 2syl 14 . . . . . . . 8 (EXMID → (∀𝑦𝑥 (𝑓𝑦) = 1o ∨ ¬ ∀𝑦𝑥 (𝑓𝑦) = 1o))
43orcomd 730 . . . . . . 7 (EXMID → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1o ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o))
54adantr 276 . . . . . 6 ((EXMID𝑓:𝑥⟶2o) → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1o ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o))
6 ffvelcdm 5669 . . . . . . . . . . . . . 14 ((𝑓:𝑥⟶2o𝑦𝑥) → (𝑓𝑦) ∈ 2o)
7 df2o3 6454 . . . . . . . . . . . . . 14 2o = {∅, 1o}
86, 7eleqtrdi 2282 . . . . . . . . . . . . 13 ((𝑓:𝑥⟶2o𝑦𝑥) → (𝑓𝑦) ∈ {∅, 1o})
9 elpri 3630 . . . . . . . . . . . . 13 ((𝑓𝑦) ∈ {∅, 1o} → ((𝑓𝑦) = ∅ ∨ (𝑓𝑦) = 1o))
108, 9syl 14 . . . . . . . . . . . 12 ((𝑓:𝑥⟶2o𝑦𝑥) → ((𝑓𝑦) = ∅ ∨ (𝑓𝑦) = 1o))
1110ord 725 . . . . . . . . . . 11 ((𝑓:𝑥⟶2o𝑦𝑥) → (¬ (𝑓𝑦) = ∅ → (𝑓𝑦) = 1o))
1211ralimdva 2557 . . . . . . . . . 10 (𝑓:𝑥⟶2o → (∀𝑦𝑥 ¬ (𝑓𝑦) = ∅ → ∀𝑦𝑥 (𝑓𝑦) = 1o))
1312con3d 632 . . . . . . . . 9 (𝑓:𝑥⟶2o → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1o → ¬ ∀𝑦𝑥 ¬ (𝑓𝑦) = ∅))
1413adantl 277 . . . . . . . 8 ((EXMID𝑓:𝑥⟶2o) → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1o → ¬ ∀𝑦𝑥 ¬ (𝑓𝑦) = ∅))
15 exmidexmid 4214 . . . . . . . . . 10 (EXMIDDECID𝑦𝑥 (𝑓𝑦) = ∅)
16 dfrex2dc 2481 . . . . . . . . . 10 (DECID𝑦𝑥 (𝑓𝑦) = ∅ → (∃𝑦𝑥 (𝑓𝑦) = ∅ ↔ ¬ ∀𝑦𝑥 ¬ (𝑓𝑦) = ∅))
1715, 16syl 14 . . . . . . . . 9 (EXMID → (∃𝑦𝑥 (𝑓𝑦) = ∅ ↔ ¬ ∀𝑦𝑥 ¬ (𝑓𝑦) = ∅))
1817adantr 276 . . . . . . . 8 ((EXMID𝑓:𝑥⟶2o) → (∃𝑦𝑥 (𝑓𝑦) = ∅ ↔ ¬ ∀𝑦𝑥 ¬ (𝑓𝑦) = ∅))
1914, 18sylibrd 169 . . . . . . 7 ((EXMID𝑓:𝑥⟶2o) → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1o → ∃𝑦𝑥 (𝑓𝑦) = ∅))
2019orim1d 788 . . . . . 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 115 . . . 4 (EXMID → (𝑓:𝑥⟶2o → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o)))
2322alrimiv 1885 . . 3 (EXMID → ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o)))
24 isomni 7163 . . . 4 (𝑥 ∈ V → (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o))))
2524elv 2756 . . 3 (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o)))
2623, 25sylibr 134 . 2 (EXMID𝑥 ∈ Omni)
2726alrimiv 1885 1 (EXMID → ∀𝑥 𝑥 ∈ Omni)
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 2752  c0 3437  {cpr 3608  EXMIDwem 4212  wf 5231  cfv 5235  1oc1o 6433  2oc2o 6434  Omnicomni 7161
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-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-exmid 4213  df-id 4311  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-1o 6440  df-2o 6441  df-omni 7162
This theorem is referenced by:  exmidomni  7169
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