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Theorem exmidomniim 6795
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 6796. (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 4031 . . . . . . . . 9 (EXMIDDECID𝑦𝑥 (𝑓𝑦) = 1𝑜)
2 exmiddc 782 . . . . . . . . 9 (DECID𝑦𝑥 (𝑓𝑦) = 1𝑜 → (∀𝑦𝑥 (𝑓𝑦) = 1𝑜 ∨ ¬ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜))
31, 2syl 14 . . . . . . . 8 (EXMID → (∀𝑦𝑥 (𝑓𝑦) = 1𝑜 ∨ ¬ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜))
43orcomd 683 . . . . . . 7 (EXMID → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜 ∨ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜))
54adantr 270 . . . . . 6 ((EXMID𝑓:𝑥⟶2𝑜) → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜 ∨ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜))
6 ffvelrn 5432 . . . . . . . . . . . . . 14 ((𝑓:𝑥⟶2𝑜𝑦𝑥) → (𝑓𝑦) ∈ 2𝑜)
7 df2o3 6195 . . . . . . . . . . . . . 14 2𝑜 = {∅, 1𝑜}
86, 7syl6eleq 2180 . . . . . . . . . . . . 13 ((𝑓:𝑥⟶2𝑜𝑦𝑥) → (𝑓𝑦) ∈ {∅, 1𝑜})
9 elpri 3469 . . . . . . . . . . . . 13 ((𝑓𝑦) ∈ {∅, 1𝑜} → ((𝑓𝑦) = ∅ ∨ (𝑓𝑦) = 1𝑜))
108, 9syl 14 . . . . . . . . . . . 12 ((𝑓:𝑥⟶2𝑜𝑦𝑥) → ((𝑓𝑦) = ∅ ∨ (𝑓𝑦) = 1𝑜))
1110ord 678 . . . . . . . . . . 11 ((𝑓:𝑥⟶2𝑜𝑦𝑥) → (¬ (𝑓𝑦) = ∅ → (𝑓𝑦) = 1𝑜))
1211ralimdva 2441 . . . . . . . . . 10 (𝑓:𝑥⟶2𝑜 → (∀𝑦𝑥 ¬ (𝑓𝑦) = ∅ → ∀𝑦𝑥 (𝑓𝑦) = 1𝑜))
1312con3d 596 . . . . . . . . 9 (𝑓:𝑥⟶2𝑜 → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜 → ¬ ∀𝑦𝑥 ¬ (𝑓𝑦) = ∅))
1413adantl 271 . . . . . . . 8 ((EXMID𝑓:𝑥⟶2𝑜) → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜 → ¬ ∀𝑦𝑥 ¬ (𝑓𝑦) = ∅))
15 exmidexmid 4031 . . . . . . . . . 10 (EXMIDDECID𝑦𝑥 (𝑓𝑦) = ∅)
16 dfrex2dc 2371 . . . . . . . . . 10 (DECID𝑦𝑥 (𝑓𝑦) = ∅ → (∃𝑦𝑥 (𝑓𝑦) = ∅ ↔ ¬ ∀𝑦𝑥 ¬ (𝑓𝑦) = ∅))
1715, 16syl 14 . . . . . . . . 9 (EXMID → (∃𝑦𝑥 (𝑓𝑦) = ∅ ↔ ¬ ∀𝑦𝑥 ¬ (𝑓𝑦) = ∅))
1817adantr 270 . . . . . . . 8 ((EXMID𝑓:𝑥⟶2𝑜) → (∃𝑦𝑥 (𝑓𝑦) = ∅ ↔ ¬ ∀𝑦𝑥 ¬ (𝑓𝑦) = ∅))
1914, 18sylibrd 167 . . . . . . 7 ((EXMID𝑓:𝑥⟶2𝑜) → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜 → ∃𝑦𝑥 (𝑓𝑦) = ∅))
2019orim1d 736 . . . . . 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 113 . . . 4 (EXMID → (𝑓:𝑥⟶2𝑜 → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜)))
2322alrimiv 1802 . . 3 (EXMID → ∀𝑓(𝑓:𝑥⟶2𝑜 → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜)))
24 vex 2622 . . . 4 𝑥 ∈ V
25 isomni 6790 . . . 4 (𝑥 ∈ V → (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2𝑜 → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜))))
2624, 25ax-mp 7 . . 3 (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2𝑜 → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜)))
2723, 26sylibr 132 . 2 (EXMID𝑥 ∈ Omni)
2827alrimiv 1802 1 (EXMID → ∀𝑥 𝑥 ∈ Omni)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 664  DECID wdc 780  wal 1287   = wceq 1289  wcel 1438  wral 2359  wrex 2360  Vcvv 2619  c0 3286  {cpr 3447  EXMIDwem 4029  wf 5011  cfv 5015  1𝑜c1o 6174  2𝑜c2o 6175  Omnicomni 6786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-exmid 4030  df-id 4120  df-suc 4198  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-1o 6181  df-2o 6182  df-omni 6788
This theorem is referenced by:  exmidomni  6796
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