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Theorem rpnnen 14327
Description: The cardinality of the continuum is the same as the powerset of  om. This is a stronger statement than ruc 14343, which only asserts that  RR is uncountable, i.e. has a cardinality larger than  om. The main proof is in two parts, rpnnen1 11323 and rpnnen2 14326, each showing an injection in one direction, and this last part uses sbth 7717 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 6321 . . . . . 6  |-  ( m  =  n  ->  (
m  /  k )  =  ( n  / 
k ) )
21breq1d 4425 . . . . 5  |-  ( m  =  n  ->  (
( m  /  k
)  <  x  <->  ( n  /  k )  < 
x ) )
32cbvrabv 3055 . . . 4  |-  { m  e.  ZZ  |  ( m  /  k )  < 
x }  =  {
n  e.  ZZ  | 
( n  /  k
)  <  x }
4 oveq2 6322 . . . . . . . . . . 11  |-  ( j  =  k  ->  (
m  /  j )  =  ( m  / 
k ) )
54breq1d 4425 . . . . . . . . . 10  |-  ( j  =  k  ->  (
( m  /  j
)  <  y  <->  ( m  /  k )  < 
y ) )
65rabbidv 3047 . . . . . . . . 9  |-  ( j  =  k  ->  { m  e.  ZZ  |  ( m  /  j )  < 
y }  =  {
m  e.  ZZ  | 
( m  /  k
)  <  y }
)
76supeq1d 7985 . . . . . . . 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 6332 . . . . . . 7  |-  ( j  =  k  ->  ( sup ( { m  e.  ZZ  |  ( m  /  j )  < 
y } ,  RR ,  <  )  /  j
)  =  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
y } ,  RR ,  <  )  /  k
) )
109cbvmptv 4508 . . . . . 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 4419 . . . . . . . . . 10  |-  ( y  =  x  ->  (
( m  /  k
)  <  y  <->  ( m  /  k )  < 
x ) )
1211rabbidv 3047 . . . . . . . . 9  |-  ( y  =  x  ->  { m  e.  ZZ  |  ( m  /  k )  < 
y }  =  {
m  e.  ZZ  | 
( m  /  k
)  <  x }
)
1312supeq1d 7985 . . . . . . . 8  |-  ( y  =  x  ->  sup ( { m  e.  ZZ  |  ( m  / 
k )  <  y } ,  RR ,  <  )  =  sup ( { m  e.  ZZ  |  ( m  / 
k )  <  x } ,  RR ,  <  ) )
1413oveq1d 6329 . . . . . . 7  |-  ( y  =  x  ->  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
y } ,  RR ,  <  )  /  k
)  =  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
x } ,  RR ,  <  )  /  k
) )
1514mpteq2dv 4503 . . . . . 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 2507 . . . . 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 4508 . . . 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 11323 . . 3  |-  RR  ~<_  ( QQ 
^m  NN )
19 qnnen 14314 . . . . . . 7  |-  QQ  ~~  NN
20 nnex 10642 . . . . . . . 8  |-  NN  e.  _V
2120canth2 7750 . . . . . . 7  |-  NN  ~<  ~P NN
22 ensdomtr 7733 . . . . . . 7  |-  ( ( QQ  ~~  NN  /\  NN  ~<  ~P NN )  ->  QQ  ~<  ~P NN )
2319, 21, 22mp2an 683 . . . . . 6  |-  QQ  ~<  ~P NN
24 sdomdom 7622 . . . . . 6  |-  ( QQ 
~<  ~P NN  ->  QQ  ~<_  ~P NN )
25 mapdom1 7762 . . . . . 6  |-  ( QQ  ~<_  ~P NN  ->  ( QQ  ^m  NN )  ~<_  ( ~P NN  ^m  NN ) )
2623, 24, 25mp2b 10 . . . . 5  |-  ( QQ 
^m  NN )  ~<_  ( ~P NN  ^m  NN )
2720pw2en 7704 . . . . . 6  |-  ~P NN  ~~  ( 2o  ^m  NN )
2820enref 7627 . . . . . 6  |-  NN  ~~  NN
29 mapen 7761 . . . . . 6  |-  ( ( ~P NN  ~~  ( 2o  ^m  NN )  /\  NN  ~~  NN )  -> 
( ~P NN  ^m  NN )  ~~  ( ( 2o  ^m  NN )  ^m  NN ) )
3027, 28, 29mp2an 683 . . . . 5  |-  ( ~P NN  ^m  NN ) 
~~  ( ( 2o 
^m  NN )  ^m  NN )
31 domentr 7653 . . . . 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 683 . . . 4  |-  ( QQ 
^m  NN )  ~<_  ( ( 2o  ^m  NN )  ^m  NN )
33 2onn 7366 . . . . . . 7  |-  2o  e.  om
34 mapxpen 7763 . . . . . . 7  |-  ( ( 2o  e.  om  /\  NN  e.  _V  /\  NN  e.  _V )  ->  (
( 2o  ^m  NN )  ^m  NN )  ~~  ( 2o  ^m  ( NN  X.  NN ) ) )
3533, 20, 20, 34mp3an 1373 . . . . . 6  |-  ( ( 2o  ^m  NN )  ^m  NN )  ~~  ( 2o  ^m  ( NN  X.  NN ) )
3633elexi 3066 . . . . . . . 8  |-  2o  e.  _V
3736enref 7627 . . . . . . 7  |-  2o  ~~  2o
38 xpnnen 14311 . . . . . . 7  |-  ( NN 
X.  NN )  ~~  NN
39 mapen 7761 . . . . . . 7  |-  ( ( 2o  ~~  2o  /\  ( NN  X.  NN )  ~~  NN )  -> 
( 2o  ^m  ( NN  X.  NN ) ) 
~~  ( 2o  ^m  NN ) )
4037, 38, 39mp2an 683 . . . . . 6  |-  ( 2o 
^m  ( NN  X.  NN ) )  ~~  ( 2o  ^m  NN )
4135, 40entri 7648 . . . . 5  |-  ( ( 2o  ^m  NN )  ^m  NN )  ~~  ( 2o  ^m  NN )
4241, 27entr4i 7651 . . . 4  |-  ( ( 2o  ^m  NN )  ^m  NN )  ~~  ~P NN
43 domentr 7653 . . . 4  |-  ( ( ( QQ  ^m  NN )  ~<_  ( ( 2o 
^m  NN )  ^m  NN )  /\  (
( 2o  ^m  NN )  ^m  NN )  ~~  ~P NN )  ->  ( QQ  ^m  NN )  ~<_  ~P NN )
4432, 42, 43mp2an 683 . . 3  |-  ( QQ 
^m  NN )  ~<_  ~P NN
45 domtr 7647 . . 3  |-  ( ( RR  ~<_  ( QQ  ^m  NN )  /\  ( QQ  ^m  NN )  ~<_  ~P NN )  ->  RR  ~<_  ~P NN )
4618, 44, 45mp2an 683 . 2  |-  RR  ~<_  ~P NN
47 elequ2 1911 . . . . . . . 8  |-  ( y  =  x  ->  (
n  e.  y  <->  n  e.  x ) )
4847ifbid 3914 . . . . . . 7  |-  ( y  =  x  ->  if ( n  e.  y ,  ( ( 1  /  3 ) ^
n ) ,  0 )  =  if ( n  e.  x ,  ( ( 1  / 
3 ) ^ n
) ,  0 ) )
4948mpteq2dv 4503 . . . . . 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 1904 . . . . . . . 8  |-  ( n  =  k  ->  (
n  e.  x  <->  k  e.  x ) )
51 oveq2 6322 . . . . . . . 8  |-  ( n  =  k  ->  (
( 1  /  3
) ^ n )  =  ( ( 1  /  3 ) ^
k ) )
5250, 51ifbieq1d 3915 . . . . . . 7  |-  ( n  =  k  ->  if ( n  e.  x ,  ( ( 1  /  3 ) ^
n ) ,  0 )  =  if ( k  e.  x ,  ( ( 1  / 
3 ) ^ k
) ,  0 ) )
5352cbvmptv 4508 . . . . . 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 2511 . . . . 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 4508 . . . 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 14326 . . 3  |-  ~P NN  ~<_  ( 0 [,] 1
)
57 reex 9655 . . . 4  |-  RR  e.  _V
58 unitssre 11807 . . . 4  |-  ( 0 [,] 1 )  C_  RR
59 ssdomg 7640 . . . 4  |-  ( RR  e.  _V  ->  (
( 0 [,] 1
)  C_  RR  ->  ( 0 [,] 1 )  ~<_  RR ) )
6057, 58, 59mp2 9 . . 3  |-  ( 0 [,] 1 )  ~<_  RR
61 domtr 7647 . . 3  |-  ( ( ~P NN  ~<_  ( 0 [,] 1 )  /\  ( 0 [,] 1
)  ~<_  RR )  ->  ~P NN  ~<_  RR )
6256, 60, 61mp2an 683 . 2  |-  ~P NN  ~<_  RR
63 sbth 7717 . 2  |-  ( ( RR  ~<_  ~P NN  /\  ~P NN 
~<_  RR )  ->  RR  ~~ 
~P NN )
6446, 62, 63mp2an 683 1  |-  RR  ~~  ~P NN
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1897   {crab 2752   _Vcvv 3056    C_ wss 3415   ifcif 3892   ~Pcpw 3962   class class class wbr 4415    |-> cmpt 4474    X. cxp 4850  (class class class)co 6314   omcom 6718   2oc2o 7201    ^m cmap 7497    ~~ cen 7591    ~<_ cdom 7592    ~< csdm 7593   supcsup 7979   RRcr 9563   0cc0 9564   1c1 9565    < clt 9700    / cdiv 10296   NNcn 10636   3c3 10687   ZZcz 10965   QQcq 11292   [,]cicc 11666   ^cexp 12303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641  ax-pre-sup 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-2o 7208  df-oadd 7211  df-omul 7212  df-er 7388  df-map 7499  df-pm 7500  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-sup 7981  df-inf 7982  df-oi 8050  df-card 8398  df-acn 8401  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-div 10297  df-nn 10637  df-2 10695  df-3 10696  df-n0 10898  df-z 10966  df-uz 11188  df-q 11293  df-rp 11331  df-ico 11669  df-icc 11670  df-fz 11813  df-fzo 11946  df-fl 12059  df-seq 12245  df-exp 12304  df-hash 12547  df-cj 13210  df-re 13211  df-im 13212  df-sqrt 13346  df-abs 13347  df-limsup 13574  df-clim 13600  df-rlim 13601  df-sum 13801
This theorem is referenced by:  rexpen  14328  cpnnen  14329  rucALT  14330  cnso  14347  2ndcredom  20513  opnreen  21897
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