Theorem exmidpw2en 6970

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 4209	. . . . 5 |- ~P x e. _V
2		pp0ex 4219	. . . . . . 7 |- { (/) , { (/) } } e. _V
3		vex 2763	. . . . . . 7 |- x e. _V
4	2, 3	mapval 6716	. . . . . 6 |- ( { (/) , { (/) } } ^m x ) = { f | f : x --> { (/) , { (/) } } }
5		mapex 6710	. . . . . . 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 2266	. . . . 5 |- ( { (/) , { (/) } } ^m x ) e. _V
8	3	a1i 9	. . . . . 6 |- (EXMID -> x e. _V )
9		0ex 4157	. . . . . . 7 |- (/) e. _V
10	9	a1i 9	. . . . . 6 |- (EXMID -> (/) e. _V )
11		p0ex 4218	. . . . . . 7 |- { (/) } e. _V
12	11	a1i 9	. . . . . 6 |- (EXMID -> { (/) } e. _V )
13		0nep0 4195	. . . . . . 7 |- (/) =/= { (/) }
14	13	a1i 9	. . . . . 6 |- (EXMID -> (/) =/= { (/) } )
15		exmidexmid 4226	. . . . . . . 8 |- (EXMID -> DECID p e. q )
16	15	ralrimivw 2568	. . . . . . 7 |- (EXMID -> A. q e. ~P xDECID p e. q )
17	16	ralrimivw 2568	. . . . . 6 |- (EXMID -> A. p e. x A. q e. ~P xDECID p e. q )
18		eqid 2193	. . . . . 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 6893	. . . . 5 |- (EXMID -> ( y e. ~P x |-> ( z e. x |-> if ( z e. y , { (/) } , (/) ) ) ) : ~P x -1-1-onto-> ( { (/) , { (/) } } ^m x ) )
20		f1oen2g 6811	. . . . 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 6486	. . . . 5 |- 2o = { (/) , { (/) } }
23	22	oveq1i 5929	. . . 4 |- ( 2o ^m x ) = ( { (/) , { (/) } } ^m x )
24	21, 23	breqtrrdi 4072	. . 3 |- (EXMID -> ~P x ~~ ( 2o ^m x ) )
25	24	alrimiv 1885	. 2 |- (EXMID -> A. x ~P x ~~ ( 2o ^m x ) )
26		1oex 6479	. . . . 5 |- 1o e. _V
27		pweq 3605	. . . . . 6 |- ( x = 1o -> ~P x = ~P 1o )
28		oveq2 5927	. . . . . 6 |- ( x = 1o -> ( 2o ^m x ) = ( 2o ^m 1o ) )
29	27, 28	breq12d 4043	. . . . 5 |- ( x = 1o -> ( ~P x ~~ ( 2o ^m x ) <-> ~P 1o ~~ ( 2o ^m 1o ) ) )
30	26, 29	spcv 2855	. . . 4 |- ( A. x ~P x ~~ ( 2o ^m x ) -> ~P 1o ~~ ( 2o ^m 1o ) )
31		df1o2 6484	. . . . . 6 |- 1o = { (/) }
32	31	oveq2i 5930	. . . . 5 |- ( 2o ^m 1o ) = ( 2o ^m { (/) } )
33	22, 2	eqeltri 2266	. . . . . 6 |- 2o e. _V
34	33, 9	mapsnen 6867	. . . . 5 |- ( 2o ^m { (/) } ) ~~ 2o
35	32, 34	eqbrtri 4051	. . . 4 |- ( 2o ^m 1o ) ~~ 2o
36		entr 6840	. . . 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 6966	. . 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 2164   {cab 2179   =/= wne 2364   A.wral 2472   _Vcvv 2760   (/)c0 3447   ifcif 3558   ~Pcpw 3602   {csn 3619   {cpr 3620   class class class wbr 4030   |-> cmpt 4091  EXMIDwem 4224   -->wf 5251   -1-1-onto->wf1o 5254  (class class class)co 5919   1oc1o 6464   2oc2o 6465   ^m cmap 6704   ~~ cen 6794

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-exmid 4225  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-1o 6471  df-2o 6472  df-er 6589  df-map 6706  df-en 6797

This theorem is referenced by: (None)