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Theorem exmidomni 7068
 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 7067 . 2 (EXMID → ∀𝑥 𝑥 ∈ Omni)
2 vex 2715 . . . . . . . . . 10 𝑢 ∈ V
3 eleq1w 2218 . . . . . . . . . 10 (𝑥 = 𝑢 → (𝑥 ∈ Omni ↔ 𝑢 ∈ Omni))
42, 3spcv 2806 . . . . . . . . 9 (∀𝑥 𝑥 ∈ Omni → 𝑢 ∈ Omni)
5 xpeq1 4597 . . . . . . . . . . . . . 14 (𝑥 = 𝑢 → (𝑥 × {∅}) = (𝑢 × {∅}))
65fveq1d 5467 . . . . . . . . . . . . 13 (𝑥 = 𝑢 → ((𝑥 × {∅})‘𝑦) = ((𝑢 × {∅})‘𝑦))
76eqeq1d 2166 . . . . . . . . . . . 12 (𝑥 = 𝑢 → (((𝑥 × {∅})‘𝑦) = ∅ ↔ ((𝑢 × {∅})‘𝑦) = ∅))
87rexeqbi1dv 2661 . . . . . . . . . . 11 (𝑥 = 𝑢 → (∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ↔ ∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅))
96eqeq1d 2166 . . . . . . . . . . . 12 (𝑥 = 𝑢 → (((𝑥 × {∅})‘𝑦) = 1o ↔ ((𝑢 × {∅})‘𝑦) = 1o))
109raleqbi1dv 2660 . . . . . . . . . . 11 (𝑥 = 𝑢 → (∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1o ↔ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o))
118, 10orbi12d 783 . . . . . . . . . 10 (𝑥 = 𝑢 → ((∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1o) ↔ (∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o)))
12 vex 2715 . . . . . . . . . . . . 13 𝑥 ∈ V
13 isomni 7062 . . . . . . . . . . . . 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 4091 . . . . . . . . . . . . . 14 ∅ ∈ V
1716prid1 3665 . . . . . . . . . . . . 13 ∅ ∈ {∅, 1o}
18 df2o3 6371 . . . . . . . . . . . . 13 2o = {∅, 1o}
1917, 18eleqtrri 2233 . . . . . . . . . . . 12 ∅ ∈ 2o
2019fconst6 5366 . . . . . . . . . . 11 (𝑥 × {∅}):𝑥⟶2o
21 p0ex 4148 . . . . . . . . . . . . 13 {∅} ∈ V
2212, 21xpex 4698 . . . . . . . . . . . 12 (𝑥 × {∅}) ∈ V
23 feq1 5299 . . . . . . . . . . . . 13 (𝑓 = (𝑥 × {∅}) → (𝑓:𝑥⟶2o ↔ (𝑥 × {∅}):𝑥⟶2o))
24 fveq1 5464 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑥 × {∅}) → (𝑓𝑦) = ((𝑥 × {∅})‘𝑦))
2524eqeq1d 2166 . . . . . . . . . . . . . . 15 (𝑓 = (𝑥 × {∅}) → ((𝑓𝑦) = ∅ ↔ ((𝑥 × {∅})‘𝑦) = ∅))
2625rexbidv 2458 . . . . . . . . . . . . . 14 (𝑓 = (𝑥 × {∅}) → (∃𝑦𝑥 (𝑓𝑦) = ∅ ↔ ∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅))
2724eqeq1d 2166 . . . . . . . . . . . . . . 15 (𝑓 = (𝑥 × {∅}) → ((𝑓𝑦) = 1o ↔ ((𝑥 × {∅})‘𝑦) = 1o))
2827ralbidv 2457 . . . . . . . . . . . . . 14 (𝑓 = (𝑥 × {∅}) → (∀𝑦𝑥 (𝑓𝑦) = 1o ↔ ∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1o))
2926, 28orbi12d 783 . . . . . . . . . . . . 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 2806 . . . . . . . . . . 11 (∀𝑓(𝑓:𝑥⟶2o → (∃𝑦𝑥 (𝑓𝑦) = ∅ ∨ ∀𝑦𝑥 (𝑓𝑦) = 1o)) → ((𝑥 × {∅}):𝑥⟶2o → (∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1o)))
3215, 20, 31mpisyl 1426 . . . . . . . . . 10 (𝑥 ∈ Omni → (∃𝑦𝑥 ((𝑥 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑥 ((𝑥 × {∅})‘𝑦) = 1o))
3311, 32vtoclga 2778 . . . . . . . . 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 274 . . . . . . 7 ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅ ∨ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o))
36 simplr 520 . . . . . . . . . 10 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅) → 𝑢 ⊆ {∅})
37 rexm 3493 . . . . . . . . . . . 12 (∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅ → ∃𝑦 𝑦𝑢)
38 sssnm 3717 . . . . . . . . . . . 12 (∃𝑦 𝑦𝑢 → (𝑢 ⊆ {∅} ↔ 𝑢 = {∅}))
3937, 38syl 14 . . . . . . . . . . 11 (∃𝑦𝑢 ((𝑢 × {∅})‘𝑦) = ∅ → (𝑢 ⊆ {∅} ↔ 𝑢 = {∅}))
4039adantl 275 . . . . . . . . . 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 1508 . . . . . . . . . . . 12 𝑦(∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅})
44 nfra1 2488 . . . . . . . . . . . 12 𝑦𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o
4543, 44nfan 1545 . . . . . . . . . . 11 𝑦((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o)
46 nfcv 2299 . . . . . . . . . . 11 𝑦𝑢
47 nfcv 2299 . . . . . . . . . . 11 𝑦
48 1n0 6373 . . . . . . . . . . . . . 14 1o ≠ ∅
4948neii 2329 . . . . . . . . . . . . 13 ¬ 1o = ∅
50 simpr 109 . . . . . . . . . . . . . . . 16 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o)
5150r19.21bi 2545 . . . . . . . . . . . . . . 15 ((((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o) ∧ 𝑦𝑢) → ((𝑢 × {∅})‘𝑦) = 1o)
5216fvconst2 5680 . . . . . . . . . . . . . . . 16 (𝑦𝑢 → ((𝑢 × {∅})‘𝑦) = ∅)
5352adantl 275 . . . . . . . . . . . . . . 15 ((((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o) ∧ 𝑦𝑢) → ((𝑢 × {∅})‘𝑦) = ∅)
5451, 53eqtr3d 2192 . . . . . . . . . . . . . 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 654 . . . . . . . . . . . 12 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → ¬ 𝑦𝑢)
5756pm2.21d 609 . . . . . . . . . . 11 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → (𝑦𝑢𝑦 ∈ ∅))
5845, 46, 47, 57ssrd 3133 . . . . . . . . . 10 (((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) ∧ ∀𝑦𝑢 ((𝑢 × {∅})‘𝑦) = 1o) → 𝑢 ⊆ ∅)
59 ss0 3434 . . . . . . . . . 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 776 . . . . . . 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 719 . . . . 5 ((∀𝑥 𝑥 ∈ Omni ∧ 𝑢 ⊆ {∅}) → (𝑢 = ∅ ∨ 𝑢 = {∅}))
6564ex 114 . . . 4 (∀𝑥 𝑥 ∈ Omni → (𝑢 ⊆ {∅} → (𝑢 = ∅ ∨ 𝑢 = {∅})))
6665alrimiv 1854 . . 3 (∀𝑥 𝑥 ∈ Omni → ∀𝑢(𝑢 ⊆ {∅} → (𝑢 = ∅ ∨ 𝑢 = {∅})))
67 exmid01 4158 . . 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 698  ∀wal 1333   = wceq 1335  ∃wex 1472   ∈ wcel 2128  ∀wral 2435  ∃wrex 2436  Vcvv 2712   ⊆ wss 3102  ∅c0 3394  {csn 3560  {cpr 3561  EXMIDwem 4154   × cxp 4581  ⟶wf 5163  ‘cfv 5167  1oc1o 6350  2oc2o 6351  Omnicomni 7060 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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-exmid 4155  df-id 4252  df-suc 4330  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-fv 5175  df-1o 6357  df-2o 6358  df-omni 7061 This theorem is referenced by:  exmidlpo  7069
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