Theorem exmidpw2en 6982

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 <-> A. x ~P 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 4213	. . . . 5 |- ~P x e. _V
2		pp0ex 4223	. . . . . . 7 |- { (/) , { (/) } } e. _V
3		vex 2766	. . . . . . 7 |- x e. _V
4	2, 3	mapval 6728	. . . . . 6 |- ( { (/) , { (/) } } ^m x ) = { f | f : x --> { (/) , { (/) } } }
5		mapex 6722	. . . . . . 7 |- ( ( x e. _V /\ { (/) , { (/) } } e. _V ) -> { f | f : x --> { (/) , { (/) } } } e. _V )
6	3, 2, 5	mp2an 426	. . . . . 6 |- { f | f : x --> { (/) , { (/) } } } e. _V
7	4, 6	eqeltri 2269	. . . . 5 |- ( { (/) , { (/) } } ^m x ) e. _V
8	3	a1i 9	. . . . . 6 |- (EXMID -> x e. _V )
9		0ex 4161	. . . . . . 7 |- (/) e. _V
10	9	a1i 9	. . . . . 6 |- (EXMID -> (/) e. _V )
11		p0ex 4222	. . . . . . 7 |- { (/) } e. _V
12	11	a1i 9	. . . . . 6 |- (EXMID -> { (/) } e. _V )
13		0nep0 4199	. . . . . . 7 |- (/) =/= { (/) }
14	13	a1i 9	. . . . . 6 |- (EXMID -> (/) =/= { (/) } )
15		exmidexmid 4230	. . . . . . . 8 |- (EXMID -> DECID p e. q )
16	15	ralrimivw 2571	. . . . . . 7 |- (EXMID -> A. q e. ~P xDECID p e. q )
17	16	ralrimivw 2571	. . . . . 6 |- (EXMID -> A. p e. x A. q e. ~P xDECID p e. q )
18		eqid 2196	. . . . . 6 |- ( y e. ~P x |-> ( z e. x |-> if ( z e. y , { (/) } , (/) ) ) ) = ( y e. ~P x |-> ( z e. x |-> if ( z e. y , { (/) } , (/) ) ) )
19	8, 10, 12, 14, 17, 18	pw2f1odc 6905	. . . . 5 |- (EXMID -> ( y e. ~P x |-> ( z e. x |-> if ( z e. y , { (/) } , (/) ) ) ) : ~P x -1-1-onto-> ( { (/) , { (/) } } ^m x ) )
20		f1oen2g 6823	. . . . 5 |- ( ( ~P x e. _V /\ ( { (/) , { (/) } } ^m x ) e. _V /\ ( y e. ~P x |-> ( z e. x |-> if ( z e. 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 6498	. . . . 5 |- 2o = { (/) , { (/) } }
23	22	oveq1i 5935	. . . 4 |- ( 2o ^m x ) = ( { (/) , { (/) } } ^m x )
24	21, 23	breqtrrdi 4076	. . 3 |- (EXMID -> ~P x ~~ ( 2o ^m x ) )
25	24	alrimiv 1888	. 2 |- (EXMID -> A. x ~P x ~~ ( 2o ^m x ) )
26		1oex 6491	. . . . 5 |- 1o e. _V
27		pweq 3609	. . . . . 6 |- ( x = 1o -> ~P x = ~P 1o )
28		oveq2 5933	. . . . . 6 |- ( x = 1o -> ( 2o ^m x ) = ( 2o ^m 1o ) )
29	27, 28	breq12d 4047	. . . . 5 |- ( x = 1o -> ( ~P x ~~ ( 2o ^m x ) <-> ~P 1o ~~ ( 2o ^m 1o ) ) )
30	26, 29	spcv 2858	. . . 4 |- ( A. x ~P x ~~ ( 2o ^m x ) -> ~P 1o ~~ ( 2o ^m 1o ) )
31		df1o2 6496	. . . . . 6 |- 1o = { (/) }
32	31	oveq2i 5936	. . . . 5 |- ( 2o ^m 1o ) = ( 2o ^m { (/) } )
33	22, 2	eqeltri 2269	. . . . . 6 |- 2o e. _V
34	33, 9	mapsnen 6879	. . . . 5 |- ( 2o ^m { (/) } ) ~~ 2o
35	32, 34	eqbrtri 4055	. . . 4 |- ( 2o ^m 1o ) ~~ 2o
36		entr 6852	. . . 4 |- ( ( ~P 1o ~~ ( 2o ^m 1o ) /\ ( 2o ^m 1o ) ~~ 2o ) -> ~P 1o ~~ 2o )
37	30, 35, 36	sylancl 413	. . 3 |- ( A. x ~P x ~~ ( 2o ^m x ) -> ~P 1o ~~ 2o )
38		exmidpw 6978	. . 3 |- (EXMID <-> ~P 1o ~~ 2o )
39	37, 38	sylibr 134	. 2 |- ( A. x ~P x ~~ ( 2o ^m x ) -> EXMID )
40	25, 39	impbii 126	1 |- (EXMID <-> A. x ~P x ~~ ( 2o ^m x ) )

Syntax hints:   <-> wb 105  DECID wdc 835   A.wal 1362   = wceq 1364   e. wcel 2167   {cab 2182   =/= wne 2367   A.wral 2475   _Vcvv 2763   (/)c0 3451   ifcif 3562   ~Pcpw 3606   {csn 3623   {cpr 3624   class class class wbr 4034   |-> cmpt 4095  EXMIDwem 4228   -->wf 5255   -1-1-onto->wf1o 5258  (class class class)co 5925   1oc1o 6476   2oc2o 6477   ^m cmap 6716   ~~ cen 6806

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-exmid 4229  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-1o 6483  df-2o 6484  df-er 6601  df-map 6718  df-en 6809

This theorem is referenced by: (None)