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Theorem exmidpw2en 6963
Description: The power set of a set being equinumerous to set exponentiation with a base of ordinal 2o is equivalent to excluded middle. This is Metamath 100 proof #52. The forward direction uses excluded middle expressed as EXMID to show this equinumerosity.

The reverse direction is the one which establishes that power set being equinumerous to set exponentiation implies excluded middle. This resolves the question of whether we will be able to prove this equinumerosity theorem in the negative. (Contributed by Jim Kingdon, 13-Aug-2022.)

Assertion
Ref Expression
exmidpw2en (EXMID ↔ ∀𝑥𝒫 𝑥 ≈ (2o𝑚 𝑥))

Proof of Theorem exmidpw2en
Dummy variables 𝑓 𝑝 𝑞 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vpwex 4208 . . . . 5 𝒫 𝑥 ∈ V
2 pp0ex 4218 . . . . . . 7 {∅, {∅}} ∈ V
3 vex 2763 . . . . . . 7 𝑥 ∈ V
42, 3mapval 6709 . . . . . 6 ({∅, {∅}} ↑𝑚 𝑥) = {𝑓𝑓:𝑥⟶{∅, {∅}}}
5 mapex 6703 . . . . . . 7 ((𝑥 ∈ V ∧ {∅, {∅}} ∈ V) → {𝑓𝑓:𝑥⟶{∅, {∅}}} ∈ V)
63, 2, 5mp2an 426 . . . . . 6 {𝑓𝑓:𝑥⟶{∅, {∅}}} ∈ V
74, 6eqeltri 2266 . . . . 5 ({∅, {∅}} ↑𝑚 𝑥) ∈ V
83a1i 9 . . . . . 6 (EXMID𝑥 ∈ V)
9 0ex 4156 . . . . . . 7 ∅ ∈ V
109a1i 9 . . . . . 6 (EXMID → ∅ ∈ V)
11 p0ex 4217 . . . . . . 7 {∅} ∈ V
1211a1i 9 . . . . . 6 (EXMID → {∅} ∈ V)
13 0nep0 4194 . . . . . . 7 ∅ ≠ {∅}
1413a1i 9 . . . . . 6 (EXMID → ∅ ≠ {∅})
15 exmidexmid 4225 . . . . . . . 8 (EXMIDDECID 𝑝𝑞)
1615ralrimivw 2568 . . . . . . 7 (EXMID → ∀𝑞 ∈ 𝒫 𝑥DECID 𝑝𝑞)
1716ralrimivw 2568 . . . . . 6 (EXMID → ∀𝑝𝑥𝑞 ∈ 𝒫 𝑥DECID 𝑝𝑞)
18 eqid 2193 . . . . . 6 (𝑦 ∈ 𝒫 𝑥 ↦ (𝑧𝑥 ↦ if(𝑧𝑦, {∅}, ∅))) = (𝑦 ∈ 𝒫 𝑥 ↦ (𝑧𝑥 ↦ if(𝑧𝑦, {∅}, ∅)))
198, 10, 12, 14, 17, 18pw2f1odc 6886 . . . . 5 (EXMID → (𝑦 ∈ 𝒫 𝑥 ↦ (𝑧𝑥 ↦ if(𝑧𝑦, {∅}, ∅))):𝒫 𝑥1-1-onto→({∅, {∅}} ↑𝑚 𝑥))
20 f1oen2g 6804 . . . . 5 ((𝒫 𝑥 ∈ V ∧ ({∅, {∅}} ↑𝑚 𝑥) ∈ V ∧ (𝑦 ∈ 𝒫 𝑥 ↦ (𝑧𝑥 ↦ if(𝑧𝑦, {∅}, ∅))):𝒫 𝑥1-1-onto→({∅, {∅}} ↑𝑚 𝑥)) → 𝒫 𝑥 ≈ ({∅, {∅}} ↑𝑚 𝑥))
211, 7, 19, 20mp3an12i 1352 . . . 4 (EXMID → 𝒫 𝑥 ≈ ({∅, {∅}} ↑𝑚 𝑥))
22 df2o2 6479 . . . . 5 2o = {∅, {∅}}
2322oveq1i 5924 . . . 4 (2o𝑚 𝑥) = ({∅, {∅}} ↑𝑚 𝑥)
2421, 23breqtrrdi 4071 . . 3 (EXMID → 𝒫 𝑥 ≈ (2o𝑚 𝑥))
2524alrimiv 1885 . 2 (EXMID → ∀𝑥𝒫 𝑥 ≈ (2o𝑚 𝑥))
26 1oex 6472 . . . . 5 1o ∈ V
27 pweq 3604 . . . . . 6 (𝑥 = 1o → 𝒫 𝑥 = 𝒫 1o)
28 oveq2 5922 . . . . . 6 (𝑥 = 1o → (2o𝑚 𝑥) = (2o𝑚 1o))
2927, 28breq12d 4042 . . . . 5 (𝑥 = 1o → (𝒫 𝑥 ≈ (2o𝑚 𝑥) ↔ 𝒫 1o ≈ (2o𝑚 1o)))
3026, 29spcv 2854 . . . 4 (∀𝑥𝒫 𝑥 ≈ (2o𝑚 𝑥) → 𝒫 1o ≈ (2o𝑚 1o))
31 df1o2 6477 . . . . . 6 1o = {∅}
3231oveq2i 5925 . . . . 5 (2o𝑚 1o) = (2o𝑚 {∅})
3322, 2eqeltri 2266 . . . . . 6 2o ∈ V
3433, 9mapsnen 6860 . . . . 5 (2o𝑚 {∅}) ≈ 2o
3532, 34eqbrtri 4050 . . . 4 (2o𝑚 1o) ≈ 2o
36 entr 6833 . . . 4 ((𝒫 1o ≈ (2o𝑚 1o) ∧ (2o𝑚 1o) ≈ 2o) → 𝒫 1o ≈ 2o)
3730, 35, 36sylancl 413 . . 3 (∀𝑥𝒫 𝑥 ≈ (2o𝑚 𝑥) → 𝒫 1o ≈ 2o)
38 exmidpw 6959 . . 3 (EXMID ↔ 𝒫 1o ≈ 2o)
3937, 38sylibr 134 . 2 (∀𝑥𝒫 𝑥 ≈ (2o𝑚 𝑥) → EXMID)
4025, 39impbii 126 1 (EXMID ↔ ∀𝑥𝒫 𝑥 ≈ (2o𝑚 𝑥))
Colors of variables: wff set class
Syntax hints:  wb 105  DECID wdc 835  wal 1362   = wceq 1364  wcel 2164  {cab 2179  wne 2364  wral 2472  Vcvv 2760  c0 3446  ifcif 3557  𝒫 cpw 3601  {csn 3618  {cpr 3619   class class class wbr 4029  cmpt 4090  EXMIDwem 4223  wf 5246  1-1-ontowf1o 5249  (class class class)co 5914  1oc1o 6457  2oc2o 6458  𝑚 cmap 6697  cen 6787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-setind 4567
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-exmid 4224  df-id 4322  df-iord 4395  df-on 4397  df-suc 4400  df-xp 4663  df-rel 4664  df-cnv 4665  df-co 4666  df-dm 4667  df-rn 4668  df-res 4669  df-ima 4670  df-iota 5211  df-fun 5252  df-fn 5253  df-f 5254  df-f1 5255  df-fo 5256  df-f1o 5257  df-fv 5258  df-ov 5917  df-oprab 5918  df-mpo 5919  df-1st 6188  df-2nd 6189  df-1o 6464  df-2o 6465  df-er 6582  df-map 6699  df-en 6790
This theorem is referenced by: (None)
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