Theorem exmidpw2en 7185

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 4297	. . . . 5 ⊢ 𝒫 𝑥 ∈ V
2		pp0ex 4307	. . . . . . 7 ⊢ {∅, {∅}} ∈ V
3		vex 2818	. . . . . . 7 ⊢ 𝑥 ∈ V
4	2, 3	mapval 6907	. . . . . 6 ⊢ ({∅, {∅}} ↑𝑚 𝑥) = {𝑓 ∣ 𝑓:𝑥⟶{∅, {∅}}}
5		mapex 6901	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ {∅, {∅}} ∈ V) → {𝑓 ∣ 𝑓:𝑥⟶{∅, {∅}}} ∈ V)
6	3, 2, 5	mp2an 426	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝑥⟶{∅, {∅}}} ∈ V
7	4, 6	eqeltri 2307	. . . . 5 ⊢ ({∅, {∅}} ↑𝑚 𝑥) ∈ V
8	3	a1i 9	. . . . . 6 ⊢ (EXMID → 𝑥 ∈ V)
9		0ex 4242	. . . . . . 7 ⊢ ∅ ∈ V
10	9	a1i 9	. . . . . 6 ⊢ (EXMID → ∅ ∈ V)
11		p0ex 4306	. . . . . . 7 ⊢ {∅} ∈ V
12	11	a1i 9	. . . . . 6 ⊢ (EXMID → {∅} ∈ V)
13		0nep0 4283	. . . . . . 7 ⊢ ∅ ≠ {∅}
14	13	a1i 9	. . . . . 6 ⊢ (EXMID → ∅ ≠ {∅})
15		exmidexmid 4314	. . . . . . . 8 ⊢ (EXMID → DECID 𝑝 ∈ 𝑞)
16	15	ralrimivw 2618	. . . . . . 7 ⊢ (EXMID → ∀𝑞 ∈ 𝒫 𝑥DECID 𝑝 ∈ 𝑞)
17	16	ralrimivw 2618	. . . . . 6 ⊢ (EXMID → ∀𝑝 ∈ 𝑥 ∀𝑞 ∈ 𝒫 𝑥DECID 𝑝 ∈ 𝑞)
18		eqid 2234	. . . . . 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 7101	. . . . 5 ⊢ (EXMID → (𝑦 ∈ 𝒫 𝑥 ↦ (𝑧 ∈ 𝑥 ↦ if(𝑧 ∈ 𝑦, {∅}, ∅))):𝒫 𝑥–1-1-onto→({∅, {∅}} ↑𝑚 𝑥))
20		f1oen2g 7007	. . . . 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 1378	. . . 4 ⊢ (EXMID → 𝒫 𝑥 ≈ ({∅, {∅}} ↑𝑚 𝑥))
22		df2o2 6676	. . . . 5 ⊢ 2o = {∅, {∅}}
23	22	oveq1i 6068	. . . 4 ⊢ (2o ↑𝑚 𝑥) = ({∅, {∅}} ↑𝑚 𝑥)
24	21, 23	breqtrrdi 4156	. . 3 ⊢ (EXMID → 𝒫 𝑥 ≈ (2o ↑𝑚 𝑥))
25	24	alrimiv 1923	. 2 ⊢ (EXMID → ∀𝑥𝒫 𝑥 ≈ (2o ↑𝑚 𝑥))
26		1oex 6668	. . . . 5 ⊢ 1o ∈ V
27		pweq 3677	. . . . . 6 ⊢ (𝑥 = 1o → 𝒫 𝑥 = 𝒫 1o)
28		oveq2 6066	. . . . . 6 ⊢ (𝑥 = 1o → (2o ↑𝑚 𝑥) = (2o ↑𝑚 1o))
29	27, 28	breq12d 4127	. . . . 5 ⊢ (𝑥 = 1o → (𝒫 𝑥 ≈ (2o ↑𝑚 𝑥) ↔ 𝒫 1o ≈ (2o ↑𝑚 1o)))
30	26, 29	spcv 2913	. . . 4 ⊢ (∀𝑥𝒫 𝑥 ≈ (2o ↑𝑚 𝑥) → 𝒫 1o ≈ (2o ↑𝑚 1o))
31		df1o2 6674	. . . . . 6 ⊢ 1o = {∅}
32	31	oveq2i 6069	. . . . 5 ⊢ (2o ↑𝑚 1o) = (2o ↑𝑚 {∅})
33	22, 2	eqeltri 2307	. . . . . 6 ⊢ 2o ∈ V
34	33, 9	mapsnen 7066	. . . . 5 ⊢ (2o ↑𝑚 {∅}) ≈ 2o
35	32, 34	eqbrtri 4135	. . . 4 ⊢ (2o ↑𝑚 1o) ≈ 2o
36		entr 7037	. . . 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 7181	. . 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 842  ∀wal 1396   = wceq 1398   ∈ wcel 2205  {cab 2220   ≠ wne 2414  ∀wral 2522  Vcvv 2815  ∅c0 3512  ifcif 3624  𝒫 cpw 3674  {csn 3694  {cpr 3695   class class class wbr 4114   ↦ cmpt 4176  EXMIDwem 4312  ⟶wf 5353  –1-1-onto→wf1o 5356  (class class class)co 6058  1_oc1o 6653  2_oc2o 6654   ↑_𝑚 cmap 6895   ≈ cen 6986

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-exmid 4313  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-1o 6660  df-2o 6661  df-er 6780  df-map 6897  df-en 6989

This theorem is referenced by: (None)