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Theorem exmidomni 6980
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 6979 . 2 (EXMID → ∀𝑥 𝑥 ∈ Omni)
2 vex 2661 . . . . . . . . . 10 𝑢 ∈ V
3 eleq1w 2176 . . . . . . . . . 10 (𝑥 = 𝑢 → (𝑥 ∈ Omni ↔ 𝑢 ∈ Omni))
42, 3spcv 2751 . . . . . . . . 9 (∀𝑥 𝑥 ∈ Omni → 𝑢 ∈ Omni)
5 xpeq1 4521 . . . . . . . . . . . . . 14 (𝑥 = 𝑢 → (𝑥 × {∅}) = (𝑢 × {∅}))
65fveq1d 5389 . . . . . . . . . . . . 13 (𝑥 = 𝑢 → ((𝑥 × {∅})‘𝑦) = ((𝑢 × {∅})‘𝑦))
76eqeq1d 2124 . . . . . . . . . . . 12 (𝑥 = 𝑢 → (((𝑥 × {∅})‘𝑦) = ∅ ↔ ((𝑢 × {∅})‘𝑦) = ∅))
87rexeqbi1dv 2610 . . . . . . . . . . 11 (𝑥 = 𝑢 → (∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ↔ ∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅))
96eqeq1d 2124 . . . . . . . . . . . 12 (𝑥 = 𝑢 → (((𝑥 × {∅})‘𝑦) = 1o ↔ ((𝑢 × {∅})‘𝑦) = 1o))
109raleqbi1dv 2609 . . . . . . . . . . 11 (𝑥 = 𝑢 → (∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1o ↔ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o))
118, 10orbi12d 765 . . . . . . . . . 10 (𝑥 = 𝑢 → ((∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1o) ↔ (∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o)))
12 vex 2661 . . . . . . . . . . . . 13 𝑥 ∈ V
13 isomni 6974 . . . . . . . . . . . . 13 (𝑥 ∈ V → (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o))))
1412, 13ax-mp 5 . . . . . . . . . . . 12 (𝑥 ∈ Omni ↔ ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o)))
1514biimpi 119 . . . . . . . . . . 11 (𝑥 ∈ Omni → ∀𝑓(𝑓:𝑥⟶2o → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o)))
16 0ex 4023 . . . . . . . . . . . . . 14 ∅ ∈ V
1716prid1 3597 . . . . . . . . . . . . 13 ∅ ∈ {∅, 1o}
18 df2o3 6293 . . . . . . . . . . . . 13 2o = {∅, 1o}
1917, 18eleqtrri 2191 . . . . . . . . . . . 12 ∅ ∈ 2o
2019fconst6 5290 . . . . . . . . . . 11 (𝑥 × {∅}):𝑥⟶2o
21 p0ex 4080 . . . . . . . . . . . . 13 {∅} ∈ V
2212, 21xpex 4622 . . . . . . . . . . . 12 (𝑥 × {∅}) ∈ V
23 feq1 5223 . . . . . . . . . . . . 13 (𝑓 = (𝑥 × {∅}) → (𝑓:𝑥⟶2o ↔ (𝑥 × {∅}):𝑥⟶2o))
24 fveq1 5386 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑥 × {∅}) → (𝑓𝑦) = ((𝑥 × {∅})‘𝑦))
2524eqeq1d 2124 . . . . . . . . . . . . . . 15 (𝑓 = (𝑥 × {∅}) → ((𝑓𝑦) = ∅ ↔ ((𝑥 × {∅})‘𝑦) = ∅))
2625rexbidv 2413 . . . . . . . . . . . . . 14 (𝑓 = (𝑥 × {∅}) → (∃𝑦𝑥 (𝑓𝑦) = ∅ ↔ ∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅))
2724eqeq1d 2124 . . . . . . . . . . . . . . 15 (𝑓 = (𝑥 × {∅}) → ((𝑓𝑦) = 1o ↔ ((𝑥 × {∅})‘𝑦) = 1o))
2827ralbidv 2412 . . . . . . . . . . . . . 14 (𝑓 = (𝑥 × {∅}) → (∀𝑦𝑥 (𝑓𝑦) = 1o ↔ ∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1o))
2926, 28orbi12d 765 . . . . . . . . . . . . 13 (𝑓 = (𝑥 × {∅}) → ((∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o) ↔ (∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1o)))
3023, 29imbi12d 233 . . . . . . . . . . . 12 (𝑓 = (𝑥 × {∅}) → ((𝑓:𝑥⟶2o → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o)) ↔ ((𝑥 × {∅}):𝑥⟶2o → (∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1o))))
3122, 30spcv 2751 . . . . . . . . . . 11 (∀𝑓(𝑓:𝑥⟶2o → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o)) → ((𝑥 × {∅}):𝑥⟶2o → (∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1o)))
3215, 20, 31mpisyl 1405 . . . . . . . . . 10 (𝑥 ∈ Omni → (∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1o))
3311, 32vtoclga 2724 . . . . . . . . 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 272 . . . . . . 7 ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o))
36 simplr 502 . . . . . . . . . 10 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅) → 𝑢 ⊆ {∅})
37 rexm 3430 . . . . . . . . . . . 12 (∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅ → ∃𝑦 𝑦𝑢)
38 sssnm 3649 . . . . . . . . . . . 12 (∃𝑦 𝑦𝑢 → (𝑢 ⊆ {∅} ↔ 𝑢 = {∅}))
3937, 38syl 14 . . . . . . . . . . 11 (∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅ → (𝑢 ⊆ {∅} ↔ 𝑢 = {∅}))
4039adantl 273 . . . . . . . . . 10 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅) → (𝑢 ⊆ {∅} ↔ 𝑢 = {∅}))
4136, 40mpbid 146 . . . . . . . . 9 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅) → 𝑢 = {∅})
4241ex 114 . . . . . . . 8 ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅ → 𝑢 = {∅}))
43 nfv 1491 . . . . . . . . . . . 12 𝑦(∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅})
44 nfra1 2441 . . . . . . . . . . . 12 𝑦𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o
4543, 44nfan 1527 . . . . . . . . . . 11 𝑦((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o)
46 nfcv 2256 . . . . . . . . . . 11 𝑦𝑢
47 nfcv 2256 . . . . . . . . . . 11 𝑦
48 1n0 6295 . . . . . . . . . . . . . 14 1o ≠ ∅
4948neii 2285 . . . . . . . . . . . . 13 ¬ 1o = ∅
50 simpr 109 . . . . . . . . . . . . . . . 16 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o)
5150r19.21bi 2495 . . . . . . . . . . . . . . 15 ((((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o) ∧ 𝑦𝑢) → ((𝑢 × {∅})‘𝑦) = 1o)
5216fvconst2 5602 . . . . . . . . . . . . . . . 16 (𝑦𝑢 → ((𝑢 × {∅})‘𝑦) = ∅)
5352adantl 273 . . . . . . . . . . . . . . 15 ((((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o) ∧ 𝑦𝑢) → ((𝑢 × {∅})‘𝑦) = ∅)
5451, 53eqtr3d 2150 . . . . . . . . . . . . . 14 ((((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o) ∧ 𝑦𝑢) → 1o = ∅)
5554ex 114 . . . . . . . . . . . . 13 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → (𝑦𝑢 → 1o = ∅))
5649, 55mtoi 636 . . . . . . . . . . . 12 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → ¬ 𝑦𝑢)
5756pm2.21d 591 . . . . . . . . . . 11 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → (𝑦𝑢𝑦 ∈ ∅))
5845, 46, 47, 57ssrd 3070 . . . . . . . . . 10 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → 𝑢 ⊆ ∅)
59 ss0 3371 . . . . . . . . . 10 (𝑢 ⊆ ∅ → 𝑢 = ∅)
6058, 59syl 14 . . . . . . . . 9 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → 𝑢 = ∅)
6160ex 114 . . . . . . . 8 ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o𝑢 = ∅))
6242, 61orim12d 758 . . . . . . 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 701 . . . . 5 ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (𝑢 = ∅ ∨ 𝑢 = {∅}))
6564ex 114 . . . 4 (∀𝑥 𝑥 ∈ Omni → (𝑢 ⊆ {∅} → (𝑢 = ∅ ∨ 𝑢 = {∅})))
6665alrimiv 1828 . . 3 (∀𝑥 𝑥 ∈ Omni → ∀𝑢(𝑢 ⊆ {∅} → (𝑢 = ∅ ∨ 𝑢 = {∅})))
67 exmid01 4089 . . 3 (EXMID ↔ ∀𝑢(𝑢 ⊆ {∅} → (𝑢 = ∅ ∨ 𝑢 = {∅})))
6866, 67sylibr 133 . 2 (∀𝑥 𝑥 ∈ Omni → EXMID)
691, 68impbii 125 1 (EXMID ↔ ∀𝑥 𝑥 ∈ Omni)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 680  wal 1312   = wceq 1314  wex 1451  wcel 1463  wral 2391  wrex 2392  Vcvv 2658  wss 3039  c0 3331  {csn 3495  {cpr 3496  EXMIDwem 4086   × cxp 4505  wf 5087  cfv 5091  1oc1o 6272  2oc2o 6273  Omnicomni 6970
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-exmid 4087  df-id 4183  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-1o 6279  df-2o 6280  df-omni 6972
This theorem is referenced by:  exmidlpo  6981
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