Theorem exmidpw2en 7213

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 4314	. . . . 5 ⊢ 𝒫 𝑥 ∈ V
2		pp0ex 4324	. . . . . . 7 ⊢ {∅, {∅}} ∈ V
3		vex 2824	. . . . . . 7 ⊢ 𝑥 ∈ V
4	2, 3	mapval 6928	. . . . . 6 ⊢ ({∅, {∅}} ↑𝑚 𝑥) = {𝑓 ∣ 𝑓:𝑥⟶{∅, {∅}}}
5		mapex 6922	. . . . . . 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 4258	. . . . . . 7 ⊢ ∅ ∈ V
10	9	a1i 9	. . . . . 6 ⊢ (EXMID → ∅ ∈ V)
11		p0ex 4323	. . . . . . 7 ⊢ {∅} ∈ V
12	11	a1i 9	. . . . . 6 ⊢ (EXMID → {∅} ∈ V)
13		0nep0 4300	. . . . . . 7 ⊢ ∅ ≠ {∅}
14	13	a1i 9	. . . . . 6 ⊢ (EXMID → ∅ ≠ {∅})
15		exmidexmid 4331	. . . . . . . 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 7129	. . . . 5 ⊢ (EXMID → (𝑦 ∈ 𝒫 𝑥 ↦ (𝑧 ∈ 𝑥 ↦ if(𝑧 ∈ 𝑦, {∅}, ∅))):𝒫 𝑥–1-1-onto→({∅, {∅}} ↑𝑚 𝑥))
20		f1oen2g 7035	. . . . 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 6697	. . . . 5 ⊢ 2o = {∅, {∅}}
23	22	oveq1i 6089	. . . 4 ⊢ (2o ↑𝑚 𝑥) = ({∅, {∅}} ↑𝑚 𝑥)
24	21, 23	breqtrrdi 4170	. . 3 ⊢ (EXMID → 𝒫 𝑥 ≈ (2o ↑𝑚 𝑥))
25	24	alrimiv 1927	. 2 ⊢ (EXMID → ∀𝑥𝒫 𝑥 ≈ (2o ↑𝑚 𝑥))
26		1oex 6689	. . . . 5 ⊢ 1o ∈ V
27		pweq 3691	. . . . . 6 ⊢ (𝑥 = 1o → 𝒫 𝑥 = 𝒫 1o)
28		oveq2 6087	. . . . . 6 ⊢ (𝑥 = 1o → (2o ↑𝑚 𝑥) = (2o ↑𝑚 1o))
29	27, 28	breq12d 4141	. . . . 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 6695	. . . . . 6 ⊢ 1o = {∅}
32	31	oveq2i 6090	. . . . 5 ⊢ (2o ↑𝑚 1o) = (2o ↑𝑚 {∅})
33	22, 2	eqeltri 2311	. . . . . 6 ⊢ 2o ∈ V
34	33, 9	mapsnen 7094	. . . . 5 ⊢ (2o ↑𝑚 {∅}) ≈ 2o
35	32, 34	eqbrtri 4149	. . . 4 ⊢ (2o ↑𝑚 1o) ≈ 2o
36		entr 7065	. . . 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 7209	. . 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 846  ∀wal 1400   = wceq 1402   ∈ wcel 2209  {cab 2224   ≠ wne 2420  ∀wral 2528  Vcvv 2821  ∅c0 3520  ifcif 3638  𝒫 cpw 3688  {csn 3708  {cpr 3709   class class class wbr 4128   ↦ cmpt 4190  EXMIDwem 4329  ⟶wf 5371  –1-1-onto→wf1o 5374  (class class class)co 6079  1_oc1o 6674  2_oc2o 6675   ↑_𝑚 cmap 6916   ≈ cen 7014

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 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 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-exmid 4330  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-1o 6681  df-2o 6682  df-er 6801  df-map 6918  df-en 7017

This theorem is referenced by: (None)