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Theorem exmidomniim 7006
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 7007. (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 4115 . . . . . . . . 9 (EXMIDDECID𝑦𝑥 (𝑓𝑦) = 1o)
2 exmiddc 821 . . . . . . . . 9 (DECID𝑦𝑥 (𝑓𝑦) = 1o → (∀𝑦𝑥 (𝑓𝑦) = 1o ∨ ¬ ∀𝑦𝑥 (𝑓𝑦) = 1o))
31, 2syl 14 . . . . . . . 8 (EXMID → (∀𝑦𝑥 (𝑓𝑦) = 1o ∨ ¬ ∀𝑦𝑥 (𝑓𝑦) = 1o))
43orcomd 718 . . . . . . 7 (EXMID → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1o ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o))
54adantr 274 . . . . . 6 ((EXMID𝑓:𝑥⟶2o) → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1o ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o))
6 ffvelrn 5546 . . . . . . . . . . . . . 14 ((𝑓:𝑥⟶2o𝑦𝑥) → (𝑓𝑦) ∈ 2o)
7 df2o3 6320 . . . . . . . . . . . . . 14 2o = {∅, 1o}
86, 7eleqtrdi 2230 . . . . . . . . . . . . 13 ((𝑓:𝑥⟶2o𝑦𝑥) → (𝑓𝑦) ∈ {∅, 1o})
9 elpri 3545 . . . . . . . . . . . . 13 ((𝑓𝑦) ∈ {∅, 1o} → ((𝑓𝑦) = ∅ ∨ (𝑓𝑦) = 1o))
108, 9syl 14 . . . . . . . . . . . 12 ((𝑓:𝑥⟶2o𝑦𝑥) → ((𝑓𝑦) = ∅ ∨ (𝑓𝑦) = 1o))
1110ord 713 . . . . . . . . . . 11 ((𝑓:𝑥⟶2o𝑦𝑥) → (¬ (𝑓𝑦) = ∅ → (𝑓𝑦) = 1o))
1211ralimdva 2497 . . . . . . . . . 10 (𝑓:𝑥⟶2o → (∀𝑦𝑥 ¬ (𝑓𝑦) = ∅ → ∀𝑦𝑥 (𝑓𝑦) = 1o))
1312con3d 620 . . . . . . . . 9 (𝑓:𝑥⟶2o → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1o → ¬ ∀𝑦𝑥 ¬ (𝑓𝑦) = ∅))
1413adantl 275 . . . . . . . 8 ((EXMID𝑓:𝑥⟶2o) → (¬ ∀𝑦𝑥 (𝑓𝑦) = 1o → ¬ ∀𝑦𝑥 ¬ (𝑓𝑦) = ∅))
15 exmidexmid 4115 . . . . . . . . . 10 (EXMIDDECID𝑦𝑥 (𝑓𝑦) = ∅)
16 dfrex2dc 2426 . . . . . . . . . 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 776 . . . . . 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 1846 . . 3 (EXMID → ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o)))
24 isomni 7001 . . . 4 (𝑥 ∈ V → (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o))))
2524elv 2685 . . 3 (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o)))
2623, 25sylibr 133 . 2 (EXMID𝑥 ∈ Omni)
2726alrimiv 1846 1 (EXMID → ∀𝑥 𝑥 ∈ Omni)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 697  DECID wdc 819  wal 1329   = wceq 1331  wcel 1480  wral 2414  wrex 2415  Vcvv 2681  c0 3358  {cpr 3523  EXMIDwem 4113  wf 5114  cfv 5118  1oc1o 6299  2oc2o 6300  Omnicomni 6997
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-exmid 4114  df-id 4210  df-suc 4288  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-1o 6306  df-2o 6307  df-omni 6999
This theorem is referenced by:  exmidomni  7007
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