Theorem exmidpw2en 6939

Description: The power set of a set being equinumerous to set exponentiation with a base of ordinal 2_o 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 ≈ (2o ↑𝑚 𝑥))

Proof of Theorem exmidpw2en

Dummy variables 𝑓 𝑝 𝑞 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vpwex 4197	. . . . 5 ⊢ 𝒫 𝑥 ∈ V
2		pp0ex 4207	. . . . . . 7 ⊢ {∅, {∅}} ∈ V
3		vex 2755	. . . . . . 7 ⊢ 𝑥 ∈ V
4	2, 3	mapval 6685	. . . . . 6 ⊢ ({∅, {∅}} ↑𝑚 𝑥) = {𝑓 ∣ 𝑓:𝑥⟶{∅, {∅}}}
5		mapex 6679	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ {∅, {∅}} ∈ V) → {𝑓 ∣ 𝑓:𝑥⟶{∅, {∅}}} ∈ V)
6	3, 2, 5	mp2an 426	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝑥⟶{∅, {∅}}} ∈ V
7	4, 6	eqeltri 2262	. . . . 5 ⊢ ({∅, {∅}} ↑𝑚 𝑥) ∈ V
8	3	a1i 9	. . . . . 6 ⊢ (EXMID → 𝑥 ∈ V)
9		0ex 4145	. . . . . . 7 ⊢ ∅ ∈ V
10	9	a1i 9	. . . . . 6 ⊢ (EXMID → ∅ ∈ V)
11		p0ex 4206	. . . . . . 7 ⊢ {∅} ∈ V
12	11	a1i 9	. . . . . 6 ⊢ (EXMID → {∅} ∈ V)
13		0nep0 4183	. . . . . . 7 ⊢ ∅ ≠ {∅}
14	13	a1i 9	. . . . . 6 ⊢ (EXMID → ∅ ≠ {∅})
15		exmidexmid 4214	. . . . . . . 8 ⊢ (EXMID → DECID 𝑝 ∈ 𝑞)
16	15	ralrimivw 2564	. . . . . . 7 ⊢ (EXMID → ∀𝑞 ∈ 𝒫 𝑥DECID 𝑝 ∈ 𝑞)
17	16	ralrimivw 2564	. . . . . 6 ⊢ (EXMID → ∀𝑝 ∈ 𝑥 ∀𝑞 ∈ 𝒫 𝑥DECID 𝑝 ∈ 𝑞)
18		eqid 2189	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 𝑥 ↦ (𝑧 ∈ 𝑥 ↦ if(𝑧 ∈ 𝑦, {∅}, ∅))) = (𝑦 ∈ 𝒫 𝑥 ↦ (𝑧 ∈ 𝑥 ↦ if(𝑧 ∈ 𝑦, {∅}, ∅)))
19	8, 10, 12, 14, 17, 18	pw2f1odc 6862	. . . . 5 ⊢ (EXMID → (𝑦 ∈ 𝒫 𝑥 ↦ (𝑧 ∈ 𝑥 ↦ if(𝑧 ∈ 𝑦, {∅}, ∅))):𝒫 𝑥–1-1-onto→({∅, {∅}} ↑𝑚 𝑥))
20		f1oen2g 6780	. . . . 5 ⊢ ((𝒫 𝑥 ∈ V ∧ ({∅, {∅}} ↑𝑚 𝑥) ∈ V ∧ (𝑦 ∈ 𝒫 𝑥 ↦ (𝑧 ∈ 𝑥 ↦ if(𝑧 ∈ 𝑦, {∅}, ∅))):𝒫 𝑥–1-1-onto→({∅, {∅}} ↑𝑚 𝑥)) → 𝒫 𝑥 ≈ ({∅, {∅}} ↑𝑚 𝑥))
21	1, 7, 19, 20	mp3an12i 1352	. . . 4 ⊢ (EXMID → 𝒫 𝑥 ≈ ({∅, {∅}} ↑𝑚 𝑥))
22		df2o2 6455	. . . . 5 ⊢ 2o = {∅, {∅}}
23	22	oveq1i 5905	. . . 4 ⊢ (2o ↑𝑚 𝑥) = ({∅, {∅}} ↑𝑚 𝑥)
24	21, 23	breqtrrdi 4060	. . 3 ⊢ (EXMID → 𝒫 𝑥 ≈ (2o ↑𝑚 𝑥))
25	24	alrimiv 1885	. 2 ⊢ (EXMID → ∀𝑥𝒫 𝑥 ≈ (2o ↑𝑚 𝑥))
26		1oex 6448	. . . . 5 ⊢ 1o ∈ V
27		pweq 3593	. . . . . 6 ⊢ (𝑥 = 1o → 𝒫 𝑥 = 𝒫 1o)
28		oveq2 5903	. . . . . 6 ⊢ (𝑥 = 1o → (2o ↑𝑚 𝑥) = (2o ↑𝑚 1o))
29	27, 28	breq12d 4031	. . . . 5 ⊢ (𝑥 = 1o → (𝒫 𝑥 ≈ (2o ↑𝑚 𝑥) ↔ 𝒫 1o ≈ (2o ↑𝑚 1o)))
30	26, 29	spcv 2846	. . . 4 ⊢ (∀𝑥𝒫 𝑥 ≈ (2o ↑𝑚 𝑥) → 𝒫 1o ≈ (2o ↑𝑚 1o))
31		df1o2 6453	. . . . . 6 ⊢ 1o = {∅}
32	31	oveq2i 5906	. . . . 5 ⊢ (2o ↑𝑚 1o) = (2o ↑𝑚 {∅})
33	22, 2	eqeltri 2262	. . . . . 6 ⊢ 2o ∈ V
34	33, 9	mapsnen 6836	. . . . 5 ⊢ (2o ↑𝑚 {∅}) ≈ 2o
35	32, 34	eqbrtri 4039	. . . 4 ⊢ (2o ↑𝑚 1o) ≈ 2o
36		entr 6809	. . . 4 ⊢ ((𝒫 1o ≈ (2o ↑𝑚 1o) ∧ (2o ↑𝑚 1o) ≈ 2o) → 𝒫 1o ≈ 2o)
37	30, 35, 36	sylancl 413	. . 3 ⊢ (∀𝑥𝒫 𝑥 ≈ (2o ↑𝑚 𝑥) → 𝒫 1o ≈ 2o)
38		exmidpw 6935	. . 3 ⊢ (EXMID ↔ 𝒫 1o ≈ 2o)
39	37, 38	sylibr 134	. 2 ⊢ (∀𝑥𝒫 𝑥 ≈ (2o ↑𝑚 𝑥) → EXMID)
40	25, 39	impbii 126	1 ⊢ (EXMID ↔ ∀𝑥𝒫 𝑥 ≈ (2o ↑𝑚 𝑥))

Syntax hints:   ↔ wb 105  DECID wdc 835  ∀wal 1362   = wceq 1364   ∈ wcel 2160  {cab 2175   ≠ wne 2360  ∀wral 2468  Vcvv 2752  ∅c0 3437  ifcif 3549  𝒫 cpw 3590  {csn 3607  {cpr 3608   class class class wbr 4018   ↦ cmpt 4079  EXMIDwem 4212  ⟶wf 5231  –1-1-onto→wf1o 5234  (class class class)co 5895  1_oc1o 6433  2_oc2o 6434   ↑_𝑚 cmap 6673   ≈ cen 6763

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-exmid 4213  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-1o 6440  df-2o 6441  df-er 6558  df-map 6675  df-en 6766

This theorem is referenced by: (None)