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Theorem exmidomni 6703
Description: Excluded middle is equivalent to every set being omniscient. (Contributed by BJ and Jim Kingdon, 30-Jun-2022.)
Assertion
Ref Expression
exmidomni (EXMID ↔ ∀𝑥 𝑥 ∈ Omni)

Proof of Theorem exmidomni
Dummy variables 𝑢 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exmidomniim 6702 . 2 (EXMID → ∀𝑥 𝑥 ∈ Omni)
2 vex 2615 . . . . . . . . . 10 𝑢 ∈ V
3 eleq1w 2143 . . . . . . . . . 10 (𝑥 = 𝑢 → (𝑥 ∈ Omni ↔ 𝑢 ∈ Omni))
42, 3spcv 2702 . . . . . . . . 9 (∀𝑥 𝑥 ∈ Omni → 𝑢 ∈ Omni)
5 xpeq1 4415 . . . . . . . . . . . . . 14 (𝑥 = 𝑢 → (𝑥 × {∅}) = (𝑢 × {∅}))
65fveq1d 5255 . . . . . . . . . . . . 13 (𝑥 = 𝑢 → ((𝑥 × {∅})‘𝑦) = ((𝑢 × {∅})‘𝑦))
76eqeq1d 2091 . . . . . . . . . . . 12 (𝑥 = 𝑢 → (((𝑥 × {∅})‘𝑦) = ∅ ↔ ((𝑢 × {∅})‘𝑦) = ∅))
87rexeqbi1dv 2564 . . . . . . . . . . 11 (𝑥 = 𝑢 → (∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ↔ ∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅))
96eqeq1d 2091 . . . . . . . . . . . 12 (𝑥 = 𝑢 → (((𝑥 × {∅})‘𝑦) = 1𝑜 ↔ ((𝑢 × {∅})‘𝑦) = 1𝑜))
109raleqbi1dv 2563 . . . . . . . . . . 11 (𝑥 = 𝑢 → (∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1𝑜 ↔ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜))
118, 10orbi12d 740 . . . . . . . . . 10 (𝑥 = 𝑢 → ((∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1𝑜) ↔ (∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜)))
12 vex 2615 . . . . . . . . . . . . 13 𝑥 ∈ V
13 isomni 6697 . . . . . . . . . . . . 13 (𝑥 ∈ V → (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2𝑜 → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜))))
1412, 13ax-mp 7 . . . . . . . . . . . 12 (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2𝑜 → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜)))
1514biimpi 118 . . . . . . . . . . 11 (𝑥 ∈ Omni → ∀𝑓(𝑓:𝑥⟶2𝑜 → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜)))
16 0ex 3931 . . . . . . . . . . . . . 14 ∅ ∈ V
1716prid1 3522 . . . . . . . . . . . . 13 ∅ ∈ {∅, 1𝑜}
18 df2o3 6127 . . . . . . . . . . . . 13 2𝑜 = {∅, 1𝑜}
1917, 18eleqtrri 2158 . . . . . . . . . . . 12 ∅ ∈ 2𝑜
2019fconst6 5158 . . . . . . . . . . 11 (𝑥 × {∅}):𝑥⟶2𝑜
21 p0ex 3987 . . . . . . . . . . . . 13 {∅} ∈ V
2212, 21xpex 4511 . . . . . . . . . . . 12 (𝑥 × {∅}) ∈ V
23 feq1 5098 . . . . . . . . . . . . 13 (𝑓 = (𝑥 × {∅}) → (𝑓:𝑥⟶2𝑜 ↔ (𝑥 × {∅}):𝑥⟶2𝑜))
24 fveq1 5252 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑥 × {∅}) → (𝑓𝑦) = ((𝑥 × {∅})‘𝑦))
2524eqeq1d 2091 . . . . . . . . . . . . . . 15 (𝑓 = (𝑥 × {∅}) → ((𝑓𝑦) = ∅ ↔ ((𝑥 × {∅})‘𝑦) = ∅))
2625rexbidv 2375 . . . . . . . . . . . . . 14 (𝑓 = (𝑥 × {∅}) → (∃𝑦𝑥 (𝑓𝑦) = ∅ ↔ ∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅))
2724eqeq1d 2091 . . . . . . . . . . . . . . 15 (𝑓 = (𝑥 × {∅}) → ((𝑓𝑦) = 1𝑜 ↔ ((𝑥 × {∅})‘𝑦) = 1𝑜))
2827ralbidv 2374 . . . . . . . . . . . . . 14 (𝑓 = (𝑥 × {∅}) → (∀𝑦𝑥 (𝑓𝑦) = 1𝑜 ↔ ∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1𝑜))
2926, 28orbi12d 740 . . . . . . . . . . . . 13 (𝑓 = (𝑥 × {∅}) → ((∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜) ↔ (∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1𝑜)))
3023, 29imbi12d 232 . . . . . . . . . . . 12 (𝑓 = (𝑥 × {∅}) → ((𝑓:𝑥⟶2𝑜 → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜)) ↔ ((𝑥 × {∅}):𝑥⟶2𝑜 → (∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1𝑜))))
3122, 30spcv 2702 . . . . . . . . . . 11 (∀𝑓(𝑓:𝑥⟶2𝑜 → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1𝑜)) → ((𝑥 × {∅}):𝑥⟶2𝑜 → (∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1𝑜)))
3215, 20, 31mpisyl 1376 . . . . . . . . . 10 (𝑥 ∈ Omni → (∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1𝑜))
3311, 32vtoclga 2675 . . . . . . . . 9 (𝑢 ∈ Omni → (∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜))
344, 33syl 14 . . . . . . . 8 (∀𝑥 𝑥 ∈ Omni → (∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜))
3534adantr 270 . . . . . . 7 ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜))
36 simplr 497 . . . . . . . . . 10 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅) → 𝑢 ⊆ {∅})
37 rexm 3362 . . . . . . . . . . . 12 (∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅ → ∃𝑦 𝑦𝑢)
38 sssnm 3572 . . . . . . . . . . . 12 (∃𝑦 𝑦𝑢 → (𝑢 ⊆ {∅} ↔ 𝑢 = {∅}))
3937, 38syl 14 . . . . . . . . . . 11 (∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅ → (𝑢 ⊆ {∅} ↔ 𝑢 = {∅}))
4039adantl 271 . . . . . . . . . 10 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅) → (𝑢 ⊆ {∅} ↔ 𝑢 = {∅}))
4136, 40mpbid 145 . . . . . . . . 9 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅) → 𝑢 = {∅})
4241ex 113 . . . . . . . 8 ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅ → 𝑢 = {∅}))
43 nfv 1462 . . . . . . . . . . . 12 𝑦(∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅})
44 nfra1 2403 . . . . . . . . . . . 12 𝑦𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜
4543, 44nfan 1498 . . . . . . . . . . 11 𝑦((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜)
46 nfcv 2223 . . . . . . . . . . 11 𝑦𝑢
47 nfcv 2223 . . . . . . . . . . 11 𝑦
48 1n0 6129 . . . . . . . . . . . . . 14 1𝑜 ≠ ∅
4948neii 2251 . . . . . . . . . . . . 13 ¬ 1𝑜 = ∅
50 simpr 108 . . . . . . . . . . . . . . . 16 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜) → ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜)
5150r19.21bi 2455 . . . . . . . . . . . . . . 15 ((((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜) ∧ 𝑦𝑢) → ((𝑢 × {∅})‘𝑦) = 1𝑜)
5216fvconst2 5453 . . . . . . . . . . . . . . . 16 (𝑦𝑢 → ((𝑢 × {∅})‘𝑦) = ∅)
5352adantl 271 . . . . . . . . . . . . . . 15 ((((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜) ∧ 𝑦𝑢) → ((𝑢 × {∅})‘𝑦) = ∅)
5451, 53eqtr3d 2117 . . . . . . . . . . . . . 14 ((((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜) ∧ 𝑦𝑢) → 1𝑜 = ∅)
5554ex 113 . . . . . . . . . . . . 13 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜) → (𝑦𝑢 → 1𝑜 = ∅))
5649, 55mtoi 623 . . . . . . . . . . . 12 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜) → ¬ 𝑦𝑢)
5756pm2.21d 582 . . . . . . . . . . 11 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜) → (𝑦𝑢𝑦 ∈ ∅))
5845, 46, 47, 57ssrd 3015 . . . . . . . . . 10 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜) → 𝑢 ⊆ ∅)
59 ss0 3305 . . . . . . . . . 10 (𝑢 ⊆ ∅ → 𝑢 = ∅)
6058, 59syl 14 . . . . . . . . 9 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜) → 𝑢 = ∅)
6160ex 113 . . . . . . . 8 ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜𝑢 = ∅))
6242, 61orim12d 733 . . . . . . 7 ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → ((∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1𝑜) → (𝑢 = {∅} ∨ 𝑢 = ∅)))
6335, 62mpd 13 . . . . . 6 ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (𝑢 = {∅} ∨ 𝑢 = ∅))
6463orcomd 681 . . . . 5 ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (𝑢 = ∅ ∨ 𝑢 = {∅}))
6564ex 113 . . . 4 (∀𝑥 𝑥 ∈ Omni → (𝑢 ⊆ {∅} → (𝑢 = ∅ ∨ 𝑢 = {∅})))
6665alrimiv 1797 . . 3 (∀𝑥 𝑥 ∈ Omni → ∀𝑢(𝑢 ⊆ {∅} → (𝑢 = ∅ ∨ 𝑢 = {∅})))
67 exmid01 3996 . . 3 (EXMID ↔ ∀𝑢(𝑢 ⊆ {∅} → (𝑢 = ∅ ∨ 𝑢 = {∅})))
6866, 67sylibr 132 . 2 (∀𝑥 𝑥 ∈ Omni → EXMID)
691, 68impbii 124 1 (EXMID ↔ ∀𝑥 𝑥 ∈ Omni)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662  wal 1283   = wceq 1285  wex 1422  wcel 1434  wral 2353  wrex 2354  Vcvv 2612  wss 2984  c0 3269  {csn 3422  {cpr 3423  EXMIDwem 3993   × cxp 4399  wf 4965  cfv 4969  1𝑜c1o 6106  2𝑜c2o 6107  Omnicomni 6695
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-mpt 3867  df-exmid 3994  df-id 4084  df-suc 4162  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-fv 4977  df-1o 6113  df-2o 6114  df-omni 6696
This theorem is referenced by: (None)
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