Theorem exmidpw2en 7099

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 4267	. . . . 5 |- ~P x e. _V
2		pp0ex 4277	. . . . . . 7 |- { (/) , { (/) } } e. _V
3		vex 2803	. . . . . . 7 |- x e. _V
4	2, 3	mapval 6824	. . . . . 6 |- ( { (/) , { (/) } } ^m x ) = { f | f : x --> { (/) , { (/) } } }
5		mapex 6818	. . . . . . 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 2302	. . . . 5 |- ( { (/) , { (/) } } ^m x ) e. _V
8	3	a1i 9	. . . . . 6 |- (EXMID -> x e. _V )
9		0ex 4214	. . . . . . 7 |- (/) e. _V
10	9	a1i 9	. . . . . 6 |- (EXMID -> (/) e. _V )
11		p0ex 4276	. . . . . . 7 |- { (/) } e. _V
12	11	a1i 9	. . . . . 6 |- (EXMID -> { (/) } e. _V )
13		0nep0 4253	. . . . . . 7 |- (/) =/= { (/) }
14	13	a1i 9	. . . . . 6 |- (EXMID -> (/) =/= { (/) } )
15		exmidexmid 4284	. . . . . . . 8 |- (EXMID -> DECID p e. q )
16	15	ralrimivw 2604	. . . . . . 7 |- (EXMID -> A. q e. ~P xDECID p e. q )
17	16	ralrimivw 2604	. . . . . 6 |- (EXMID -> A. p e. x A. q e. ~P xDECID p e. q )
18		eqid 2229	. . . . . 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 7016	. . . . 5 |- (EXMID -> ( y e. ~P x |-> ( z e. x |-> if ( z e. y , { (/) } , (/) ) ) ) : ~P x -1-1-onto-> ( { (/) , { (/) } } ^m x ) )
20		f1oen2g 6923	. . . . 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 1375	. . . 4 |- (EXMID -> ~P x ~~ ( { (/) , { (/) } } ^m x ) )
22		df2o2 6593	. . . . 5 |- 2o = { (/) , { (/) } }
23	22	oveq1i 6023	. . . 4 |- ( 2o ^m x ) = ( { (/) , { (/) } } ^m x )
24	21, 23	breqtrrdi 4128	. . 3 |- (EXMID -> ~P x ~~ ( 2o ^m x ) )
25	24	alrimiv 1920	. 2 |- (EXMID -> A. x ~P x ~~ ( 2o ^m x ) )
26		1oex 6585	. . . . 5 |- 1o e. _V
27		pweq 3653	. . . . . 6 |- ( x = 1o -> ~P x = ~P 1o )
28		oveq2 6021	. . . . . 6 |- ( x = 1o -> ( 2o ^m x ) = ( 2o ^m 1o ) )
29	27, 28	breq12d 4099	. . . . 5 |- ( x = 1o -> ( ~P x ~~ ( 2o ^m x ) <-> ~P 1o ~~ ( 2o ^m 1o ) ) )
30	26, 29	spcv 2898	. . . 4 |- ( A. x ~P x ~~ ( 2o ^m x ) -> ~P 1o ~~ ( 2o ^m 1o ) )
31		df1o2 6591	. . . . . 6 |- 1o = { (/) }
32	31	oveq2i 6024	. . . . 5 |- ( 2o ^m 1o ) = ( 2o ^m { (/) } )
33	22, 2	eqeltri 2302	. . . . . 6 |- 2o e. _V
34	33, 9	mapsnen 6981	. . . . 5 |- ( 2o ^m { (/) } ) ~~ 2o
35	32, 34	eqbrtri 4107	. . . 4 |- ( 2o ^m 1o ) ~~ 2o
36		entr 6953	. . . 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 7095	. . 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 839   A.wal 1393   = wceq 1395   e. wcel 2200   {cab 2215   =/= wne 2400   A.wral 2508   _Vcvv 2800   (/)c0 3492   ifcif 3603   ~Pcpw 3650   {csn 3667   {cpr 3668   class class class wbr 4086   |-> cmpt 4148  EXMIDwem 4282   -->wf 5320   -1-1-onto->wf1o 5323  (class class class)co 6013   1oc1o 6570   2oc2o 6571   ^m cmap 6812   ~~ cen 6902

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-exmid 4283  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-1o 6577  df-2o 6578  df-er 6697  df-map 6814  df-en 6905

This theorem is referenced by: (None)