Theorem exmidpw2en 7224

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 4318	. . . . 5 ⊢ 𝒫 𝑥 ∈ V
2		pp0ex 4328	. . . . . . 7 ⊢ {∅, {∅}} ∈ V
3		vex 2824	. . . . . . 7 ⊢ 𝑥 ∈ V
4	2, 3	mapval 6938	. . . . . 6 ⊢ ({∅, {∅}} ↑𝑚 𝑥) = {𝑓 ∣ 𝑓:𝑥⟶{∅, {∅}}}
5		mapex 6932	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ {∅, {∅}} ∈ V) → {𝑓 ∣ 𝑓:𝑥⟶{∅, {∅}}} ∈ V)
6	3, 2, 5	mp2an 430	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝑥⟶{∅, {∅}}} ∈ V
7	4, 6	eqeltri 2311	. . . . 5 ⊢ ({∅, {∅}} ↑𝑚 𝑥) ∈ V
8	3	a1i 9	. . . . . 6 ⊢ (EXMID → 𝑥 ∈ V)
9		0ex 4262	. . . . . . 7 ⊢ ∅ ∈ V
10	9	a1i 9	. . . . . 6 ⊢ (EXMID → ∅ ∈ V)
11		p0ex 4327	. . . . . . 7 ⊢ {∅} ∈ V
12	11	a1i 9	. . . . . 6 ⊢ (EXMID → {∅} ∈ V)
13		0nep0 4304	. . . . . . 7 ⊢ ∅ ≠ {∅}
14	13	a1i 9	. . . . . 6 ⊢ (EXMID → ∅ ≠ {∅})
15		exmidexmid 4335	. . . . . . . 8 ⊢ (EXMID → DECID 𝑝 ∈ 𝑞)
16	15	ralrimivw 2624	. . . . . . 7 ⊢ (EXMID → ∀𝑞 ∈ 𝒫 𝑥DECID 𝑝 ∈ 𝑞)
17	16	ralrimivw 2624	. . . . . 6 ⊢ (EXMID → ∀𝑝 ∈ 𝑥 ∀𝑞 ∈ 𝒫 𝑥DECID 𝑝 ∈ 𝑞)
18		eqid 2238	. . . . . 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 7140	. . . . 5 ⊢ (EXMID → (𝑦 ∈ 𝒫 𝑥 ↦ (𝑧 ∈ 𝑥 ↦ if(𝑧 ∈ 𝑦, {∅}, ∅))):𝒫 𝑥–1-1-onto→({∅, {∅}} ↑𝑚 𝑥))
20		f1oen2g 7046	. . . . 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 1382	. . . 4 ⊢ (EXMID → 𝒫 𝑥 ≈ ({∅, {∅}} ↑𝑚 𝑥))
22		df2o2 6707	. . . . 5 ⊢ 2o = {∅, {∅}}
23	22	oveq1i 6098	. . . 4 ⊢ (2o ↑𝑚 𝑥) = ({∅, {∅}} ↑𝑚 𝑥)
24	21, 23	breqtrrdi 4173	. . 3 ⊢ (EXMID → 𝒫 𝑥 ≈ (2o ↑𝑚 𝑥))
25	24	alrimiv 1927	. 2 ⊢ (EXMID → ∀𝑥𝒫 𝑥 ≈ (2o ↑𝑚 𝑥))
26		1oex 6699	. . . . 5 ⊢ 1o ∈ V
27		pweq 3691	. . . . . 6 ⊢ (𝑥 = 1o → 𝒫 𝑥 = 𝒫 1o)
28		oveq2 6096	. . . . . 6 ⊢ (𝑥 = 1o → (2o ↑𝑚 𝑥) = (2o ↑𝑚 1o))
29	27, 28	breq12d 4144	. . . . 5 ⊢ (𝑥 = 1o → (𝒫 𝑥 ≈ (2o ↑𝑚 𝑥) ↔ 𝒫 1o ≈ (2o ↑𝑚 1o)))
30	26, 29	spcv 2919	. . . 4 ⊢ (∀𝑥𝒫 𝑥 ≈ (2o ↑𝑚 𝑥) → 𝒫 1o ≈ (2o ↑𝑚 1o))
31		df1o2 6705	. . . . . 6 ⊢ 1o = {∅}
32	31	oveq2i 6099	. . . . 5 ⊢ (2o ↑𝑚 1o) = (2o ↑𝑚 {∅})
33	22, 2	eqeltri 2311	. . . . . 6 ⊢ 2o ∈ V
34	33, 9	mapsnen 7105	. . . . 5 ⊢ (2o ↑𝑚 {∅}) ≈ 2o
35	32, 34	eqbrtri 4152	. . . 4 ⊢ (2o ↑𝑚 1o) ≈ 2o
36		entr 7076	. . . 4 ⊢ ((𝒫 1o ≈ (2o ↑𝑚 1o) ∧ (2o ↑𝑚 1o) ≈ 2o) → 𝒫 1o ≈ 2o)
37	30, 35, 36	sylancl 417	. . 3 ⊢ (∀𝑥𝒫 𝑥 ≈ (2o ↑𝑚 𝑥) → 𝒫 1o ≈ 2o)
38		exmidpw 7220	. . 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))

This proof depends on syntax axioms:   ↔ wb 105  DECID wdc 846  ∀wal 1400   = wceq 1402   ∈ wcel 2209  {cab 2224   ≠ wne 2420  ∀wral 2528  Vcvv 2821  ∅c0 3520  ifcif 3638  𝒫 cpw 3688  {csn 3709  {cpr 3710   class class class wbr 4131   ↦ cmpt 4193  EXMIDwem 4333  ⟶wf 5376  –1-1-onto→wf1o 5379  (class class class)co 6088  1_oc1o 6684  2_oc2o 6685   ↑_𝑚 cmap 6926   ≈ cen 7025

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-exmid 4334  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-1o 6691  df-2o 6692  df-er 6811  df-map 6928  df-en 7028

This theorem is used by: (None)