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Theorem rpnnen 12781
Description: The cardinality of the continuum is the same as the powerset of  om. This is a stronger statement than ruc 12797, which only asserts that  RR is uncountable, i.e. has a cardinality larger than  om. The main proof is in two parts, rpnnen1 10561 and rpnnen2 12780, each showing an injection in one direction, and this last part uses sbth 7186 to prove that the sets are equinumerous. By constructing explicit injections, we avoid the use of AC. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 23-Aug-2014.)
Assertion
Ref Expression
rpnnen  |-  RR  ~~  ~P NN

Proof of Theorem rpnnen
Dummy variables  j 
k  m  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6047 . . . . . 6  |-  ( m  =  n  ->  (
m  /  k )  =  ( n  / 
k ) )
21breq1d 4182 . . . . 5  |-  ( m  =  n  ->  (
( m  /  k
)  <  x  <->  ( n  /  k )  < 
x ) )
32cbvrabv 2915 . . . 4  |-  { m  e.  ZZ  |  ( m  /  k )  < 
x }  =  {
n  e.  ZZ  | 
( n  /  k
)  <  x }
4 oveq2 6048 . . . . . . . . . . 11  |-  ( j  =  k  ->  (
m  /  j )  =  ( m  / 
k ) )
54breq1d 4182 . . . . . . . . . 10  |-  ( j  =  k  ->  (
( m  /  j
)  <  y  <->  ( m  /  k )  < 
y ) )
65rabbidv 2908 . . . . . . . . 9  |-  ( j  =  k  ->  { m  e.  ZZ  |  ( m  /  j )  < 
y }  =  {
m  e.  ZZ  | 
( m  /  k
)  <  y }
)
76supeq1d 7409 . . . . . . . 8  |-  ( j  =  k  ->  sup ( { m  e.  ZZ  |  ( m  / 
j )  <  y } ,  RR ,  <  )  =  sup ( { m  e.  ZZ  |  ( m  / 
k )  <  y } ,  RR ,  <  ) )
8 id 20 . . . . . . . 8  |-  ( j  =  k  ->  j  =  k )
97, 8oveq12d 6058 . . . . . . 7  |-  ( j  =  k  ->  ( sup ( { m  e.  ZZ  |  ( m  /  j )  < 
y } ,  RR ,  <  )  /  j
)  =  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
y } ,  RR ,  <  )  /  k
) )
109cbvmptv 4260 . . . . . 6  |-  ( j  e.  NN  |->  ( sup ( { m  e.  ZZ  |  ( m  /  j )  < 
y } ,  RR ,  <  )  /  j
) )  =  ( k  e.  NN  |->  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
y } ,  RR ,  <  )  /  k
) )
11 breq2 4176 . . . . . . . . . 10  |-  ( y  =  x  ->  (
( m  /  k
)  <  y  <->  ( m  /  k )  < 
x ) )
1211rabbidv 2908 . . . . . . . . 9  |-  ( y  =  x  ->  { m  e.  ZZ  |  ( m  /  k )  < 
y }  =  {
m  e.  ZZ  | 
( m  /  k
)  <  x }
)
1312supeq1d 7409 . . . . . . . 8  |-  ( y  =  x  ->  sup ( { m  e.  ZZ  |  ( m  / 
k )  <  y } ,  RR ,  <  )  =  sup ( { m  e.  ZZ  |  ( m  / 
k )  <  x } ,  RR ,  <  ) )
1413oveq1d 6055 . . . . . . 7  |-  ( y  =  x  ->  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
y } ,  RR ,  <  )  /  k
)  =  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
x } ,  RR ,  <  )  /  k
) )
1514mpteq2dv 4256 . . . . . 6  |-  ( y  =  x  ->  (
k  e.  NN  |->  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
y } ,  RR ,  <  )  /  k
) )  =  ( k  e.  NN  |->  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
x } ,  RR ,  <  )  /  k
) ) )
1610, 15syl5eq 2448 . . . . 5  |-  ( y  =  x  ->  (
j  e.  NN  |->  ( sup ( { m  e.  ZZ  |  ( m  /  j )  < 
y } ,  RR ,  <  )  /  j
) )  =  ( k  e.  NN  |->  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
x } ,  RR ,  <  )  /  k
) ) )
1716cbvmptv 4260 . . . 4  |-  ( y  e.  RR  |->  ( j  e.  NN  |->  ( sup ( { m  e.  ZZ  |  ( m  /  j )  < 
y } ,  RR ,  <  )  /  j
) ) )  =  ( x  e.  RR  |->  ( k  e.  NN  |->  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
x } ,  RR ,  <  )  /  k
) ) )
183, 17rpnnen1 10561 . . 3  |-  RR  ~<_  ( QQ 
^m  NN )
19 qnnen 12768 . . . . . . 7  |-  QQ  ~~  NN
20 nnex 9962 . . . . . . . 8  |-  NN  e.  _V
2120canth2 7219 . . . . . . 7  |-  NN  ~<  ~P NN
22 ensdomtr 7202 . . . . . . 7  |-  ( ( QQ  ~~  NN  /\  NN  ~<  ~P NN )  ->  QQ  ~<  ~P NN )
2319, 21, 22mp2an 654 . . . . . 6  |-  QQ  ~<  ~P NN
24 sdomdom 7094 . . . . . 6  |-  ( QQ 
~<  ~P NN  ->  QQ  ~<_  ~P NN )
25 mapdom1 7231 . . . . . 6  |-  ( QQ  ~<_  ~P NN  ->  ( QQ  ^m  NN )  ~<_  ( ~P NN  ^m  NN ) )
2623, 24, 25mp2b 10 . . . . 5  |-  ( QQ 
^m  NN )  ~<_  ( ~P NN  ^m  NN )
2720pw2en 7174 . . . . . 6  |-  ~P NN  ~~  ( 2o  ^m  NN )
2820enref 7099 . . . . . 6  |-  NN  ~~  NN
29 mapen 7230 . . . . . 6  |-  ( ( ~P NN  ~~  ( 2o  ^m  NN )  /\  NN  ~~  NN )  -> 
( ~P NN  ^m  NN )  ~~  ( ( 2o  ^m  NN )  ^m  NN ) )
3027, 28, 29mp2an 654 . . . . 5  |-  ( ~P NN  ^m  NN ) 
~~  ( ( 2o 
^m  NN )  ^m  NN )
31 domentr 7125 . . . . 5  |-  ( ( ( QQ  ^m  NN )  ~<_  ( ~P NN  ^m  NN )  /\  ( ~P NN  ^m  NN ) 
~~  ( ( 2o 
^m  NN )  ^m  NN ) )  ->  ( QQ  ^m  NN )  ~<_  ( ( 2o  ^m  NN )  ^m  NN ) )
3226, 30, 31mp2an 654 . . . 4  |-  ( QQ 
^m  NN )  ~<_  ( ( 2o  ^m  NN )  ^m  NN )
33 2onn 6842 . . . . . . 7  |-  2o  e.  om
34 mapxpen 7232 . . . . . . 7  |-  ( ( 2o  e.  om  /\  NN  e.  _V  /\  NN  e.  _V )  ->  (
( 2o  ^m  NN )  ^m  NN )  ~~  ( 2o  ^m  ( NN  X.  NN ) ) )
3533, 20, 20, 34mp3an 1279 . . . . . 6  |-  ( ( 2o  ^m  NN )  ^m  NN )  ~~  ( 2o  ^m  ( NN  X.  NN ) )
3633elexi 2925 . . . . . . . 8  |-  2o  e.  _V
3736enref 7099 . . . . . . 7  |-  2o  ~~  2o
38 xpnnen 12763 . . . . . . 7  |-  ( NN 
X.  NN )  ~~  NN
39 mapen 7230 . . . . . . 7  |-  ( ( 2o  ~~  2o  /\  ( NN  X.  NN )  ~~  NN )  -> 
( 2o  ^m  ( NN  X.  NN ) ) 
~~  ( 2o  ^m  NN ) )
4037, 38, 39mp2an 654 . . . . . 6  |-  ( 2o 
^m  ( NN  X.  NN ) )  ~~  ( 2o  ^m  NN )
4135, 40entri 7120 . . . . 5  |-  ( ( 2o  ^m  NN )  ^m  NN )  ~~  ( 2o  ^m  NN )
4241, 27entr4i 7123 . . . 4  |-  ( ( 2o  ^m  NN )  ^m  NN )  ~~  ~P NN
43 domentr 7125 . . . 4  |-  ( ( ( QQ  ^m  NN )  ~<_  ( ( 2o 
^m  NN )  ^m  NN )  /\  (
( 2o  ^m  NN )  ^m  NN )  ~~  ~P NN )  ->  ( QQ  ^m  NN )  ~<_  ~P NN )
4432, 42, 43mp2an 654 . . 3  |-  ( QQ 
^m  NN )  ~<_  ~P NN
45 domtr 7119 . . 3  |-  ( ( RR  ~<_  ( QQ  ^m  NN )  /\  ( QQ  ^m  NN )  ~<_  ~P NN )  ->  RR  ~<_  ~P NN )
4618, 44, 45mp2an 654 . 2  |-  RR  ~<_  ~P NN
47 elequ2 1726 . . . . . . . 8  |-  ( y  =  x  ->  (
n  e.  y  <->  n  e.  x ) )
4847ifbid 3717 . . . . . . 7  |-  ( y  =  x  ->  if ( n  e.  y ,  ( ( 1  /  3 ) ^
n ) ,  0 )  =  if ( n  e.  x ,  ( ( 1  / 
3 ) ^ n
) ,  0 ) )
4948mpteq2dv 4256 . . . . . 6  |-  ( y  =  x  ->  (
n  e.  NN  |->  if ( n  e.  y ,  ( ( 1  /  3 ) ^
n ) ,  0 ) )  =  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  /  3 ) ^
n ) ,  0 ) ) )
50 elequ1 1724 . . . . . . . 8  |-  ( n  =  k  ->  (
n  e.  x  <->  k  e.  x ) )
51 oveq2 6048 . . . . . . . 8  |-  ( n  =  k  ->  (
( 1  /  3
) ^ n )  =  ( ( 1  /  3 ) ^
k ) )
52 eqidd 2405 . . . . . . . 8  |-  ( n  =  k  ->  0  =  0 )
5350, 51, 52ifbieq12d 3721 . . . . . . 7  |-  ( n  =  k  ->  if ( n  e.  x ,  ( ( 1  /  3 ) ^
n ) ,  0 )  =  if ( k  e.  x ,  ( ( 1  / 
3 ) ^ k
) ,  0 ) )
5453cbvmptv 4260 . . . . . 6  |-  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
3 ) ^ n
) ,  0 ) )  =  ( k  e.  NN  |->  if ( k  e.  x ,  ( ( 1  / 
3 ) ^ k
) ,  0 ) )
5549, 54syl6eq 2452 . . . . 5  |-  ( y  =  x  ->  (
n  e.  NN  |->  if ( n  e.  y ,  ( ( 1  /  3 ) ^
n ) ,  0 ) )  =  ( k  e.  NN  |->  if ( k  e.  x ,  ( ( 1  /  3 ) ^
k ) ,  0 ) ) )
5655cbvmptv 4260 . . . 4  |-  ( y  e.  ~P NN  |->  ( n  e.  NN  |->  if ( n  e.  y ,  ( ( 1  /  3 ) ^
n ) ,  0 ) ) )  =  ( x  e.  ~P NN  |->  ( k  e.  NN  |->  if ( k  e.  x ,  ( ( 1  /  3
) ^ k ) ,  0 ) ) )
5756rpnnen2 12780 . . 3  |-  ~P NN  ~<_  ( 0 [,] 1
)
58 reex 9037 . . . 4  |-  RR  e.  _V
59 unitssre 10998 . . . 4  |-  ( 0 [,] 1 )  C_  RR
60 ssdomg 7112 . . . 4  |-  ( RR  e.  _V  ->  (
( 0 [,] 1
)  C_  RR  ->  ( 0 [,] 1 )  ~<_  RR ) )
6158, 59, 60mp2 9 . . 3  |-  ( 0 [,] 1 )  ~<_  RR
62 domtr 7119 . . 3  |-  ( ( ~P NN  ~<_  ( 0 [,] 1 )  /\  ( 0 [,] 1
)  ~<_  RR )  ->  ~P NN  ~<_  RR )
6357, 61, 62mp2an 654 . 2  |-  ~P NN  ~<_  RR
64 sbth 7186 . 2  |-  ( ( RR  ~<_  ~P NN  /\  ~P NN 
~<_  RR )  ->  RR  ~~ 
~P NN )
6546, 63, 64mp2an 654 1  |-  RR  ~~  ~P NN
Colors of variables: wff set class
Syntax hints:    e. wcel 1721   {crab 2670   _Vcvv 2916    C_ wss 3280   ifcif 3699   ~Pcpw 3759   class class class wbr 4172    e. cmpt 4226   omcom 4804    X. cxp 4835  (class class class)co 6040   2oc2o 6677    ^m cmap 6977    ~~ cen 7065    ~<_ cdom 7066    ~< csdm 7067   supcsup 7403   RRcr 8945   0cc0 8946   1c1 8947    < clt 9076    / cdiv 9633   NNcn 9956   3c3 10006   ZZcz 10238   QQcq 10530   [,]cicc 10875   ^cexp 11337
This theorem is referenced by:  rexpen  12782  cpnnen  12783  rucALT  12784  cnso  12801  2ndcredom  17466  opnreen  18815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435
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