Theorem exmidpw2en 7174

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 4294	. . . . 5 |- ~P x e. _V
2		pp0ex 4304	. . . . . . 7 |- { (/) , { (/) } } e. _V
3		vex 2818	. . . . . . 7 |- x e. _V
4	2, 3	mapval 6896	. . . . . 6 |- ( { (/) , { (/) } } ^m x ) = { f | f : x --> { (/) , { (/) } } }
5		mapex 6890	. . . . . . 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 2307	. . . . 5 |- ( { (/) , { (/) } } ^m x ) e. _V
8	3	a1i 9	. . . . . 6 |- (EXMID -> x e. _V )
9		0ex 4239	. . . . . . 7 |- (/) e. _V
10	9	a1i 9	. . . . . 6 |- (EXMID -> (/) e. _V )
11		p0ex 4303	. . . . . . 7 |- { (/) } e. _V
12	11	a1i 9	. . . . . 6 |- (EXMID -> { (/) } e. _V )
13		0nep0 4280	. . . . . . 7 |- (/) =/= { (/) }
14	13	a1i 9	. . . . . 6 |- (EXMID -> (/) =/= { (/) } )
15		exmidexmid 4311	. . . . . . . 8 |- (EXMID -> DECID p e. q )
16	15	ralrimivw 2618	. . . . . . 7 |- (EXMID -> A. q e. ~P xDECID p e. q )
17	16	ralrimivw 2618	. . . . . 6 |- (EXMID -> A. p e. x A. q e. ~P xDECID p e. q )
18		eqid 2234	. . . . . 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 7090	. . . . 5 |- (EXMID -> ( y e. ~P x |-> ( z e. x |-> if ( z e. y , { (/) } , (/) ) ) ) : ~P x -1-1-onto-> ( { (/) , { (/) } } ^m x ) )
20		f1oen2g 6996	. . . . 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 1378	. . . 4 |- (EXMID -> ~P x ~~ ( { (/) , { (/) } } ^m x ) )
22		df2o2 6665	. . . . 5 |- 2o = { (/) , { (/) } }
23	22	oveq1i 6062	. . . 4 |- ( 2o ^m x ) = ( { (/) , { (/) } } ^m x )
24	21, 23	breqtrrdi 4153	. . 3 |- (EXMID -> ~P x ~~ ( 2o ^m x ) )
25	24	alrimiv 1923	. 2 |- (EXMID -> A. x ~P x ~~ ( 2o ^m x ) )
26		1oex 6657	. . . . 5 |- 1o e. _V
27		pweq 3674	. . . . . 6 |- ( x = 1o -> ~P x = ~P 1o )
28		oveq2 6060	. . . . . 6 |- ( x = 1o -> ( 2o ^m x ) = ( 2o ^m 1o ) )
29	27, 28	breq12d 4124	. . . . 5 |- ( x = 1o -> ( ~P x ~~ ( 2o ^m x ) <-> ~P 1o ~~ ( 2o ^m 1o ) ) )
30	26, 29	spcv 2913	. . . 4 |- ( A. x ~P x ~~ ( 2o ^m x ) -> ~P 1o ~~ ( 2o ^m 1o ) )
31		df1o2 6663	. . . . . 6 |- 1o = { (/) }
32	31	oveq2i 6063	. . . . 5 |- ( 2o ^m 1o ) = ( 2o ^m { (/) } )
33	22, 2	eqeltri 2307	. . . . . 6 |- 2o e. _V
34	33, 9	mapsnen 7055	. . . . 5 |- ( 2o ^m { (/) } ) ~~ 2o
35	32, 34	eqbrtri 4132	. . . 4 |- ( 2o ^m 1o ) ~~ 2o
36		entr 7026	. . . 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 7170	. . 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 842   A.wal 1396   = wceq 1398   e. wcel 2205   {cab 2220   =/= wne 2414   A.wral 2522   _Vcvv 2815   (/)c0 3510   ifcif 3622   ~Pcpw 3671   {csn 3691   {cpr 3692   class class class wbr 4111   |-> cmpt 4173  EXMIDwem 4309   -->wf 5350   -1-1-onto->wf1o 5353  (class class class)co 6052   1oc1o 6642   2oc2o 6643   ^m cmap 6884   ~~ cen 6975

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-exmid 4310  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-1o 6649  df-2o 6650  df-er 6769  df-map 6886  df-en 6978

This theorem is referenced by: (None)