Theorem exmidpw2en 7219

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 4316	. . . . 5 |- ~P x e. _V
2		pp0ex 4326	. . . . . . 7 |- { (/) , { (/) } } e. _V
3		vex 2824	. . . . . . 7 |- x e. _V
4	2, 3	mapval 6934	. . . . . 6 |- ( { (/) , { (/) } } ^m x ) = { f | f : x --> { (/) , { (/) } } }
5		mapex 6928	. . . . . . 7 |- ( ( x e. _V /\ { (/) , { (/) } } e. _V ) -> { f | f : x --> { (/) , { (/) } } } e. _V )
6	3, 2, 5	mp2an 430	. . . . . 6 |- { f | f : x --> { (/) , { (/) } } } e. _V
7	4, 6	eqeltri 2311	. . . . 5 |- ( { (/) , { (/) } } ^m x ) e. _V
8	3	a1i 9	. . . . . 6 |- (EXMID -> x e. _V )
9		0ex 4260	. . . . . . 7 |- (/) e. _V
10	9	a1i 9	. . . . . 6 |- (EXMID -> (/) e. _V )
11		p0ex 4325	. . . . . . 7 |- { (/) } e. _V
12	11	a1i 9	. . . . . 6 |- (EXMID -> { (/) } e. _V )
13		0nep0 4302	. . . . . . 7 |- (/) =/= { (/) }
14	13	a1i 9	. . . . . 6 |- (EXMID -> (/) =/= { (/) } )
15		exmidexmid 4333	. . . . . . . 8 |- (EXMID -> DECID p e. q )
16	15	ralrimivw 2624	. . . . . . 7 |- (EXMID -> A. q e. ~P xDECID p e. q )
17	16	ralrimivw 2624	. . . . . 6 |- (EXMID -> A. p e. x A. q e. ~P xDECID p e. q )
18		eqid 2238	. . . . . 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 7135	. . . . 5 |- (EXMID -> ( y e. ~P x |-> ( z e. x |-> if ( z e. y , { (/) } , (/) ) ) ) : ~P x -1-1-onto-> ( { (/) , { (/) } } ^m x ) )
20		f1oen2g 7041	. . . . 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 1382	. . . 4 |- (EXMID -> ~P x ~~ ( { (/) , { (/) } } ^m x ) )
22		df2o2 6703	. . . . 5 |- 2o = { (/) , { (/) } }
23	22	oveq1i 6095	. . . 4 |- ( 2o ^m x ) = ( { (/) , { (/) } } ^m x )
24	21, 23	breqtrrdi 4172	. . 3 |- (EXMID -> ~P x ~~ ( 2o ^m x ) )
25	24	alrimiv 1927	. 2 |- (EXMID -> A. x ~P x ~~ ( 2o ^m x ) )
26		1oex 6695	. . . . 5 |- 1o e. _V
27		pweq 3691	. . . . . 6 |- ( x = 1o -> ~P x = ~P 1o )
28		oveq2 6093	. . . . . 6 |- ( x = 1o -> ( 2o ^m x ) = ( 2o ^m 1o ) )
29	27, 28	breq12d 4143	. . . . 5 |- ( x = 1o -> ( ~P x ~~ ( 2o ^m x ) <-> ~P 1o ~~ ( 2o ^m 1o ) ) )
30	26, 29	spcv 2919	. . . 4 |- ( A. x ~P x ~~ ( 2o ^m x ) -> ~P 1o ~~ ( 2o ^m 1o ) )
31		df1o2 6701	. . . . . 6 |- 1o = { (/) }
32	31	oveq2i 6096	. . . . 5 |- ( 2o ^m 1o ) = ( 2o ^m { (/) } )
33	22, 2	eqeltri 2311	. . . . . 6 |- 2o e. _V
34	33, 9	mapsnen 7100	. . . . 5 |- ( 2o ^m { (/) } ) ~~ 2o
35	32, 34	eqbrtri 4151	. . . 4 |- ( 2o ^m 1o ) ~~ 2o
36		entr 7071	. . . 4 |- ( ( ~P 1o ~~ ( 2o ^m 1o ) /\ ( 2o ^m 1o ) ~~ 2o ) -> ~P 1o ~~ 2o )
37	30, 35, 36	sylancl 417	. . 3 |- ( A. x ~P x ~~ ( 2o ^m x ) -> ~P 1o ~~ 2o )
38		exmidpw 7215	. . 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 ) )

This proof depends on syntax axioms:   <-> wb 105  DECID wdc 846   A.wal 1400   = wceq 1402   e. wcel 2209   {cab 2224   =/= wne 2420   A.wral 2528   _Vcvv 2821   (/)c0 3520   ifcif 3638   ~Pcpw 3688   {csn 3709   {cpr 3710   class class class wbr 4130   |-> cmpt 4192  EXMIDwem 4331   -->wf 5373   -1-1-onto->wf1o 5376  (class class class)co 6085   1oc1o 6680   2oc2o 6681   ^m cmap 6922   ~~ cen 7020

This proof depends on 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 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-exmid 4332  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-1o 6687  df-2o 6688  df-er 6807  df-map 6924  df-en 7023

This theorem is used by: (None)