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Theorem rpnnen 13812
Description: The cardinality of the continuum is the same as the powerset of  om. This is a stronger statement than ruc 13828, which only asserts that  RR is uncountable, i.e. has a cardinality larger than  om. The main proof is in two parts, rpnnen1 11204 and rpnnen2 13811, each showing an injection in one direction, and this last part uses sbth 7629 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 6284 . . . . . 6  |-  ( m  =  n  ->  (
m  /  k )  =  ( n  / 
k ) )
21breq1d 4452 . . . . 5  |-  ( m  =  n  ->  (
( m  /  k
)  <  x  <->  ( n  /  k )  < 
x ) )
32cbvrabv 3107 . . . 4  |-  { m  e.  ZZ  |  ( m  /  k )  < 
x }  =  {
n  e.  ZZ  | 
( n  /  k
)  <  x }
4 oveq2 6285 . . . . . . . . . . 11  |-  ( j  =  k  ->  (
m  /  j )  =  ( m  / 
k ) )
54breq1d 4452 . . . . . . . . . 10  |-  ( j  =  k  ->  (
( m  /  j
)  <  y  <->  ( m  /  k )  < 
y ) )
65rabbidv 3100 . . . . . . . . 9  |-  ( j  =  k  ->  { m  e.  ZZ  |  ( m  /  j )  < 
y }  =  {
m  e.  ZZ  | 
( m  /  k
)  <  y }
)
76supeq1d 7897 . . . . . . . 8  |-  ( j  =  k  ->  sup ( { m  e.  ZZ  |  ( m  / 
j )  <  y } ,  RR ,  <  )  =  sup ( { m  e.  ZZ  |  ( m  / 
k )  <  y } ,  RR ,  <  ) )
8 id 22 . . . . . . . 8  |-  ( j  =  k  ->  j  =  k )
97, 8oveq12d 6295 . . . . . . 7  |-  ( j  =  k  ->  ( sup ( { m  e.  ZZ  |  ( m  /  j )  < 
y } ,  RR ,  <  )  /  j
)  =  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
y } ,  RR ,  <  )  /  k
) )
109cbvmptv 4533 . . . . . 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 4446 . . . . . . . . . 10  |-  ( y  =  x  ->  (
( m  /  k
)  <  y  <->  ( m  /  k )  < 
x ) )
1211rabbidv 3100 . . . . . . . . 9  |-  ( y  =  x  ->  { m  e.  ZZ  |  ( m  /  k )  < 
y }  =  {
m  e.  ZZ  | 
( m  /  k
)  <  x }
)
1312supeq1d 7897 . . . . . . . 8  |-  ( y  =  x  ->  sup ( { m  e.  ZZ  |  ( m  / 
k )  <  y } ,  RR ,  <  )  =  sup ( { m  e.  ZZ  |  ( m  / 
k )  <  x } ,  RR ,  <  ) )
1413oveq1d 6292 . . . . . . 7  |-  ( y  =  x  ->  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
y } ,  RR ,  <  )  /  k
)  =  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
x } ,  RR ,  <  )  /  k
) )
1514mpteq2dv 4529 . . . . . 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 2515 . . . . 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 4533 . . . 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 11204 . . 3  |-  RR  ~<_  ( QQ 
^m  NN )
19 qnnen 13799 . . . . . . 7  |-  QQ  ~~  NN
20 nnex 10533 . . . . . . . 8  |-  NN  e.  _V
2120canth2 7662 . . . . . . 7  |-  NN  ~<  ~P NN
22 ensdomtr 7645 . . . . . . 7  |-  ( ( QQ  ~~  NN  /\  NN  ~<  ~P NN )  ->  QQ  ~<  ~P NN )
2319, 21, 22mp2an 672 . . . . . 6  |-  QQ  ~<  ~P NN
24 sdomdom 7535 . . . . . 6  |-  ( QQ 
~<  ~P NN  ->  QQ  ~<_  ~P NN )
25 mapdom1 7674 . . . . . 6  |-  ( QQ  ~<_  ~P NN  ->  ( QQ  ^m  NN )  ~<_  ( ~P NN  ^m  NN ) )
2623, 24, 25mp2b 10 . . . . 5  |-  ( QQ 
^m  NN )  ~<_  ( ~P NN  ^m  NN )
2720pw2en 7616 . . . . . 6  |-  ~P NN  ~~  ( 2o  ^m  NN )
2820enref 7540 . . . . . 6  |-  NN  ~~  NN
29 mapen 7673 . . . . . 6  |-  ( ( ~P NN  ~~  ( 2o  ^m  NN )  /\  NN  ~~  NN )  -> 
( ~P NN  ^m  NN )  ~~  ( ( 2o  ^m  NN )  ^m  NN ) )
3027, 28, 29mp2an 672 . . . . 5  |-  ( ~P NN  ^m  NN ) 
~~  ( ( 2o 
^m  NN )  ^m  NN )
31 domentr 7566 . . . . 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 672 . . . 4  |-  ( QQ 
^m  NN )  ~<_  ( ( 2o  ^m  NN )  ^m  NN )
33 2onn 7281 . . . . . . 7  |-  2o  e.  om
34 mapxpen 7675 . . . . . . 7  |-  ( ( 2o  e.  om  /\  NN  e.  _V  /\  NN  e.  _V )  ->  (
( 2o  ^m  NN )  ^m  NN )  ~~  ( 2o  ^m  ( NN  X.  NN ) ) )
3533, 20, 20, 34mp3an 1319 . . . . . 6  |-  ( ( 2o  ^m  NN )  ^m  NN )  ~~  ( 2o  ^m  ( NN  X.  NN ) )
3633elexi 3118 . . . . . . . 8  |-  2o  e.  _V
3736enref 7540 . . . . . . 7  |-  2o  ~~  2o
38 xpnnen 13794 . . . . . . 7  |-  ( NN 
X.  NN )  ~~  NN
39 mapen 7673 . . . . . . 7  |-  ( ( 2o  ~~  2o  /\  ( NN  X.  NN )  ~~  NN )  -> 
( 2o  ^m  ( NN  X.  NN ) ) 
~~  ( 2o  ^m  NN ) )
4037, 38, 39mp2an 672 . . . . . 6  |-  ( 2o 
^m  ( NN  X.  NN ) )  ~~  ( 2o  ^m  NN )
4135, 40entri 7561 . . . . 5  |-  ( ( 2o  ^m  NN )  ^m  NN )  ~~  ( 2o  ^m  NN )
4241, 27entr4i 7564 . . . 4  |-  ( ( 2o  ^m  NN )  ^m  NN )  ~~  ~P NN
43 domentr 7566 . . . 4  |-  ( ( ( QQ  ^m  NN )  ~<_  ( ( 2o 
^m  NN )  ^m  NN )  /\  (
( 2o  ^m  NN )  ^m  NN )  ~~  ~P NN )  ->  ( QQ  ^m  NN )  ~<_  ~P NN )
4432, 42, 43mp2an 672 . . 3  |-  ( QQ 
^m  NN )  ~<_  ~P NN
45 domtr 7560 . . 3  |-  ( ( RR  ~<_  ( QQ  ^m  NN )  /\  ( QQ  ^m  NN )  ~<_  ~P NN )  ->  RR  ~<_  ~P NN )
4618, 44, 45mp2an 672 . 2  |-  RR  ~<_  ~P NN
47 elequ2 1767 . . . . . . . 8  |-  ( y  =  x  ->  (
n  e.  y  <->  n  e.  x ) )
4847ifbid 3956 . . . . . . 7  |-  ( y  =  x  ->  if ( n  e.  y ,  ( ( 1  /  3 ) ^
n ) ,  0 )  =  if ( n  e.  x ,  ( ( 1  / 
3 ) ^ n
) ,  0 ) )
4948mpteq2dv 4529 . . . . . 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 1765 . . . . . . . 8  |-  ( n  =  k  ->  (
n  e.  x  <->  k  e.  x ) )
51 oveq2 6285 . . . . . . . 8  |-  ( n  =  k  ->  (
( 1  /  3
) ^ n )  =  ( ( 1  /  3 ) ^
k ) )
5250, 51ifbieq1d 3957 . . . . . . 7  |-  ( n  =  k  ->  if ( n  e.  x ,  ( ( 1  /  3 ) ^
n ) ,  0 )  =  if ( k  e.  x ,  ( ( 1  / 
3 ) ^ k
) ,  0 ) )
5352cbvmptv 4533 . . . . . 6  |-  ( n  e.  NN  |->  if ( n  e.  x ,  ( ( 1  / 
3 ) ^ n
) ,  0 ) )  =  ( k  e.  NN  |->  if ( k  e.  x ,  ( ( 1  / 
3 ) ^ k
) ,  0 ) )
5449, 53syl6eq 2519 . . . . 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 ) ) )
5554cbvmptv 4533 . . . 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 ) ) )
5655rpnnen2 13811 . . 3  |-  ~P NN  ~<_  ( 0 [,] 1
)
57 reex 9574 . . . 4  |-  RR  e.  _V
58 unitssre 11658 . . . 4  |-  ( 0 [,] 1 )  C_  RR
59 ssdomg 7553 . . . 4  |-  ( RR  e.  _V  ->  (
( 0 [,] 1
)  C_  RR  ->  ( 0 [,] 1 )  ~<_  RR ) )
6057, 58, 59mp2 9 . . 3  |-  ( 0 [,] 1 )  ~<_  RR
61 domtr 7560 . . 3  |-  ( ( ~P NN  ~<_  ( 0 [,] 1 )  /\  ( 0 [,] 1
)  ~<_  RR )  ->  ~P NN  ~<_  RR )
6256, 60, 61mp2an 672 . 2  |-  ~P NN  ~<_  RR
63 sbth 7629 . 2  |-  ( ( RR  ~<_  ~P NN  /\  ~P NN 
~<_  RR )  ->  RR  ~~ 
~P NN )
6446, 62, 63mp2an 672 1  |-  RR  ~~  ~P NN
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1762   {crab 2813   _Vcvv 3108    C_ wss 3471   ifcif 3934   ~Pcpw 4005   class class class wbr 4442    |-> cmpt 4500    X. cxp 4992  (class class class)co 6277   omcom 6673   2oc2o 7116    ^m cmap 7412    ~~ cen 7505    ~<_ cdom 7506    ~< csdm 7507   supcsup 7891   RRcr 9482   0cc0 9483   1c1 9484    < clt 9619    / cdiv 10197   NNcn 10527   3c3 10577   ZZcz 10855   QQcq 11173   [,]cicc 11523   ^cexp 12124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-omul 7127  df-er 7303  df-map 7414  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-sup 7892  df-oi 7926  df-card 8311  df-acn 8314  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-q 11174  df-rp 11212  df-ico 11526  df-icc 11527  df-fz 11664  df-fzo 11784  df-fl 11888  df-seq 12066  df-exp 12125  df-hash 12363  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-limsup 13245  df-clim 13262  df-rlim 13263  df-sum 13460
This theorem is referenced by:  rexpen  13813  cpnnen  13814  rucALT  13815  cnso  13832  2ndcredom  19712  opnreen  21066
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