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Theorem rpnnen 13611
Description: The cardinality of the continuum is the same as the powerset of  om. This is a stronger statement than ruc 13627, which only asserts that  RR is uncountable, i.e. has a cardinality larger than  om. The main proof is in two parts, rpnnen1 11085 and rpnnen2 13610, each showing an injection in one direction, and this last part uses sbth 7531 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 6197 . . . . . 6  |-  ( m  =  n  ->  (
m  /  k )  =  ( n  / 
k ) )
21breq1d 4400 . . . . 5  |-  ( m  =  n  ->  (
( m  /  k
)  <  x  <->  ( n  /  k )  < 
x ) )
32cbvrabv 3067 . . . 4  |-  { m  e.  ZZ  |  ( m  /  k )  < 
x }  =  {
n  e.  ZZ  | 
( n  /  k
)  <  x }
4 oveq2 6198 . . . . . . . . . . 11  |-  ( j  =  k  ->  (
m  /  j )  =  ( m  / 
k ) )
54breq1d 4400 . . . . . . . . . 10  |-  ( j  =  k  ->  (
( m  /  j
)  <  y  <->  ( m  /  k )  < 
y ) )
65rabbidv 3060 . . . . . . . . 9  |-  ( j  =  k  ->  { m  e.  ZZ  |  ( m  /  j )  < 
y }  =  {
m  e.  ZZ  | 
( m  /  k
)  <  y }
)
76supeq1d 7797 . . . . . . . 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 6208 . . . . . . 7  |-  ( j  =  k  ->  ( sup ( { m  e.  ZZ  |  ( m  /  j )  < 
y } ,  RR ,  <  )  /  j
)  =  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
y } ,  RR ,  <  )  /  k
) )
109cbvmptv 4481 . . . . . 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 4394 . . . . . . . . . 10  |-  ( y  =  x  ->  (
( m  /  k
)  <  y  <->  ( m  /  k )  < 
x ) )
1211rabbidv 3060 . . . . . . . . 9  |-  ( y  =  x  ->  { m  e.  ZZ  |  ( m  /  k )  < 
y }  =  {
m  e.  ZZ  | 
( m  /  k
)  <  x }
)
1312supeq1d 7797 . . . . . . . 8  |-  ( y  =  x  ->  sup ( { m  e.  ZZ  |  ( m  / 
k )  <  y } ,  RR ,  <  )  =  sup ( { m  e.  ZZ  |  ( m  / 
k )  <  x } ,  RR ,  <  ) )
1413oveq1d 6205 . . . . . . 7  |-  ( y  =  x  ->  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
y } ,  RR ,  <  )  /  k
)  =  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
x } ,  RR ,  <  )  /  k
) )
1514mpteq2dv 4477 . . . . . 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 2504 . . . . 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 4481 . . . 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 11085 . . 3  |-  RR  ~<_  ( QQ 
^m  NN )
19 qnnen 13598 . . . . . . 7  |-  QQ  ~~  NN
20 nnex 10429 . . . . . . . 8  |-  NN  e.  _V
2120canth2 7564 . . . . . . 7  |-  NN  ~<  ~P NN
22 ensdomtr 7547 . . . . . . 7  |-  ( ( QQ  ~~  NN  /\  NN  ~<  ~P NN )  ->  QQ  ~<  ~P NN )
2319, 21, 22mp2an 672 . . . . . 6  |-  QQ  ~<  ~P NN
24 sdomdom 7437 . . . . . 6  |-  ( QQ 
~<  ~P NN  ->  QQ  ~<_  ~P NN )
25 mapdom1 7576 . . . . . 6  |-  ( QQ  ~<_  ~P NN  ->  ( QQ  ^m  NN )  ~<_  ( ~P NN  ^m  NN ) )
2623, 24, 25mp2b 10 . . . . 5  |-  ( QQ 
^m  NN )  ~<_  ( ~P NN  ^m  NN )
2720pw2en 7518 . . . . . 6  |-  ~P NN  ~~  ( 2o  ^m  NN )
2820enref 7442 . . . . . 6  |-  NN  ~~  NN
29 mapen 7575 . . . . . 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 7468 . . . . 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 7179 . . . . . . 7  |-  2o  e.  om
34 mapxpen 7577 . . . . . . 7  |-  ( ( 2o  e.  om  /\  NN  e.  _V  /\  NN  e.  _V )  ->  (
( 2o  ^m  NN )  ^m  NN )  ~~  ( 2o  ^m  ( NN  X.  NN ) ) )
3533, 20, 20, 34mp3an 1315 . . . . . 6  |-  ( ( 2o  ^m  NN )  ^m  NN )  ~~  ( 2o  ^m  ( NN  X.  NN ) )
3633elexi 3078 . . . . . . . 8  |-  2o  e.  _V
3736enref 7442 . . . . . . 7  |-  2o  ~~  2o
38 xpnnen 13593 . . . . . . 7  |-  ( NN 
X.  NN )  ~~  NN
39 mapen 7575 . . . . . . 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 7463 . . . . 5  |-  ( ( 2o  ^m  NN )  ^m  NN )  ~~  ( 2o  ^m  NN )
4241, 27entr4i 7466 . . . 4  |-  ( ( 2o  ^m  NN )  ^m  NN )  ~~  ~P NN
43 domentr 7468 . . . 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 7462 . . 3  |-  ( ( RR  ~<_  ( QQ  ^m  NN )  /\  ( QQ  ^m  NN )  ~<_  ~P NN )  ->  RR  ~<_  ~P NN )
4618, 44, 45mp2an 672 . 2  |-  RR  ~<_  ~P NN
47 elequ2 1763 . . . . . . . 8  |-  ( y  =  x  ->  (
n  e.  y  <->  n  e.  x ) )
4847ifbid 3909 . . . . . . 7  |-  ( y  =  x  ->  if ( n  e.  y ,  ( ( 1  /  3 ) ^
n ) ,  0 )  =  if ( n  e.  x ,  ( ( 1  / 
3 ) ^ n
) ,  0 ) )
4948mpteq2dv 4477 . . . . . 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 1761 . . . . . . . 8  |-  ( n  =  k  ->  (
n  e.  x  <->  k  e.  x ) )
51 oveq2 6198 . . . . . . . 8  |-  ( n  =  k  ->  (
( 1  /  3
) ^ n )  =  ( ( 1  /  3 ) ^
k ) )
5250, 51ifbieq1d 3910 . . . . . . 7  |-  ( n  =  k  ->  if ( n  e.  x ,  ( ( 1  /  3 ) ^
n ) ,  0 )  =  if ( k  e.  x ,  ( ( 1  / 
3 ) ^ k
) ,  0 ) )
5352cbvmptv 4481 . . . . . 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 2508 . . . . 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 4481 . . . 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 13610 . . 3  |-  ~P NN  ~<_  ( 0 [,] 1
)
57 reex 9474 . . . 4  |-  RR  e.  _V
58 unitssre 11533 . . . 4  |-  ( 0 [,] 1 )  C_  RR
59 ssdomg 7455 . . . 4  |-  ( RR  e.  _V  ->  (
( 0 [,] 1
)  C_  RR  ->  ( 0 [,] 1 )  ~<_  RR ) )
6057, 58, 59mp2 9 . . 3  |-  ( 0 [,] 1 )  ~<_  RR
61 domtr 7462 . . 3  |-  ( ( ~P NN  ~<_  ( 0 [,] 1 )  /\  ( 0 [,] 1
)  ~<_  RR )  ->  ~P NN  ~<_  RR )
6256, 60, 61mp2an 672 . 2  |-  ~P NN  ~<_  RR
63 sbth 7531 . 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 1758   {crab 2799   _Vcvv 3068    C_ wss 3426   ifcif 3889   ~Pcpw 3958   class class class wbr 4390    |-> cmpt 4448    X. cxp 4936  (class class class)co 6190   omcom 6576   2oc2o 7014    ^m cmap 7314    ~~ cen 7407    ~<_ cdom 7408    ~< csdm 7409   supcsup 7791   RRcr 9382   0cc0 9383   1c1 9384    < clt 9519    / cdiv 10094   NNcn 10423   3c3 10473   ZZcz 10747   QQcq 11054   [,]cicc 11404   ^cexp 11966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-omul 7025  df-er 7201  df-map 7316  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-sup 7792  df-oi 7825  df-card 8210  df-acn 8213  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-n0 10681  df-z 10748  df-uz 10963  df-q 11055  df-rp 11093  df-ico 11407  df-icc 11408  df-fz 11539  df-fzo 11650  df-fl 11743  df-seq 11908  df-exp 11967  df-hash 12205  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-limsup 13051  df-clim 13068  df-rlim 13069  df-sum 13266
This theorem is referenced by:  rexpen  13612  cpnnen  13613  rucALT  13614  cnso  13631  2ndcredom  19170  opnreen  20524
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