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Theorem canthp1 8278
Description: A slightly stronger form of Cantor's theorem: For  1  <  n,  n  +  1  <  2 ^ n. Corollary 1.6 of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
canthp1  |-  ( 1o 
~<  A  ->  ( A  +c  1o )  ~<  ~P A )

Proof of Theorem canthp1
Dummy variables  f 
a  g  r  s  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1sdom2 7063 . . . 4  |-  1o  ~<  2o
2 sdomdom 6891 . . . 4  |-  ( 1o 
~<  2o  ->  1o  ~<_  2o )
3 cdadom2 7815 . . . 4  |-  ( 1o  ~<_  2o  ->  ( A  +c  1o )  ~<_  ( A  +c  2o ) )
41, 2, 3mp2b 9 . . 3  |-  ( A  +c  1o )  ~<_  ( A  +c  2o )
5 canthp1lem1 8276 . . 3  |-  ( 1o 
~<  A  ->  ( A  +c  2o )  ~<_  ~P A )
6 domtr 6916 . . 3  |-  ( ( ( A  +c  1o )  ~<_  ( A  +c  2o )  /\  ( A  +c  2o )  ~<_  ~P A )  ->  ( A  +c  1o )  ~<_  ~P A )
74, 5, 6sylancr 644 . 2  |-  ( 1o 
~<  A  ->  ( A  +c  1o )  ~<_  ~P A )
8 fal 1313 . . 3  |-  -.  F.
9 ensym 6912 . . . . 5  |-  ( ( A  +c  1o ) 
~~  ~P A  ->  ~P A  ~~  ( A  +c  1o ) )
10 bren 6873 . . . . 5  |-  ( ~P A  ~~  ( A  +c  1o )  <->  E. f 
f : ~P A -1-1-onto-> ( A  +c  1o ) )
119, 10sylib 188 . . . 4  |-  ( ( A  +c  1o ) 
~~  ~P A  ->  E. f 
f : ~P A -1-1-onto-> ( A  +c  1o ) )
12 f1of 5474 . . . . . . . . . 10  |-  ( f : ~P A -1-1-onto-> ( A  +c  1o )  -> 
f : ~P A --> ( A  +c  1o ) )
13 relsdom 6872 . . . . . . . . . . . 12  |-  Rel  ~<
1413brrelex2i 4732 . . . . . . . . . . 11  |-  ( 1o 
~<  A  ->  A  e. 
_V )
15 pwidg 3639 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  A  e.  ~P A )
1614, 15syl 15 . . . . . . . . . 10  |-  ( 1o 
~<  A  ->  A  e. 
~P A )
17 ffvelrn 5665 . . . . . . . . . 10  |-  ( ( f : ~P A --> ( A  +c  1o )  /\  A  e.  ~P A )  ->  (
f `  A )  e.  ( A  +c  1o ) )
1812, 16, 17syl2anr 464 . . . . . . . . 9  |-  ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  ->  ( f `  A )  e.  ( A  +c  1o ) )
19 cda1dif 7804 . . . . . . . . 9  |-  ( ( f `  A )  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { ( f `
 A ) } )  ~~  A )
2018, 19syl 15 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  ->  ( ( A  +c  1o )  \  { ( f `  A ) } ) 
~~  A )
21 bren 6873 . . . . . . . 8  |-  ( ( ( A  +c  1o )  \  { ( f `
 A ) } )  ~~  A  <->  E. g 
g : ( ( A  +c  1o ) 
\  { ( f `
 A ) } ) -1-1-onto-> A )
2220, 21sylib 188 . . . . . . 7  |-  ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  ->  E. g  g : ( ( A  +c  1o )  \  { ( f `  A ) } ) -1-1-onto-> A )
23 simpll 730 . . . . . . . . . . 11  |-  ( ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  /\  g : ( ( A  +c  1o )  \  { ( f `
 A ) } ) -1-1-onto-> A )  ->  1o  ~<  A )
24 simplr 731 . . . . . . . . . . 11  |-  ( ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  /\  g : ( ( A  +c  1o )  \  { ( f `
 A ) } ) -1-1-onto-> A )  ->  f : ~P A -1-1-onto-> ( A  +c  1o ) )
25 simpr 447 . . . . . . . . . . 11  |-  ( ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  /\  g : ( ( A  +c  1o )  \  { ( f `
 A ) } ) -1-1-onto-> A )  ->  g : ( ( A  +c  1o )  \  { ( f `  A ) } ) -1-1-onto-> A )
26 eqeq1 2291 . . . . . . . . . . . . . 14  |-  ( w  =  x  ->  (
w  =  A  <->  x  =  A ) )
27 id 19 . . . . . . . . . . . . . 14  |-  ( w  =  x  ->  w  =  x )
2826, 27ifbieq2d 3587 . . . . . . . . . . . . 13  |-  ( w  =  x  ->  if ( w  =  A ,  (/) ,  w )  =  if ( x  =  A ,  (/) ,  x ) )
2928cbvmptv 4113 . . . . . . . . . . . 12  |-  ( w  e.  ~P A  |->  if ( w  =  A ,  (/) ,  w ) )  =  ( x  e.  ~P A  |->  if ( x  =  A ,  (/) ,  x ) )
3029coeq2i 4846 . . . . . . . . . . 11  |-  ( ( g  o.  f )  o.  ( w  e. 
~P A  |->  if ( w  =  A ,  (/)
,  w ) ) )  =  ( ( g  o.  f )  o.  ( x  e. 
~P A  |->  if ( x  =  A ,  (/)
,  x ) ) )
31 eqid 2285 . . . . . . . . . . . 12  |-  { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  ( (
( g  o.  f
)  o.  ( w  e.  ~P A  |->  if ( w  =  A ,  (/) ,  w ) ) ) `  ( `' s " {
z } ) )  =  z ) ) }  =  { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  ( (
( g  o.  f
)  o.  ( w  e.  ~P A  |->  if ( w  =  A ,  (/) ,  w ) ) ) `  ( `' s " {
z } ) )  =  z ) ) }
3231fpwwecbv 8268 . . . . . . . . . . 11  |-  { <. a ,  s >.  |  ( ( a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  ( (
( g  o.  f
)  o.  ( w  e.  ~P A  |->  if ( w  =  A ,  (/) ,  w ) ) ) `  ( `' s " {
z } ) )  =  z ) ) }  =  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( (
( g  o.  f
)  o.  ( w  e.  ~P A  |->  if ( w  =  A ,  (/) ,  w ) ) ) `  ( `' r " {
y } ) )  =  y ) ) }
33 eqid 2285 . . . . . . . . . . 11  |-  U. dom  {
<. a ,  s >.  |  ( ( a 
C_  A  /\  s  C_  ( a  X.  a
) )  /\  (
s  We  a  /\  A. z  e.  a  ( ( ( g  o.  f )  o.  (
w  e.  ~P A  |->  if ( w  =  A ,  (/) ,  w
) ) ) `  ( `' s " {
z } ) )  =  z ) ) }  =  U. dom  {
<. a ,  s >.  |  ( ( a 
C_  A  /\  s  C_  ( a  X.  a
) )  /\  (
s  We  a  /\  A. z  e.  a  ( ( ( g  o.  f )  o.  (
w  e.  ~P A  |->  if ( w  =  A ,  (/) ,  w
) ) ) `  ( `' s " {
z } ) )  =  z ) ) }
3423, 24, 25, 30, 32, 33canthp1lem2 8277 . . . . . . . . . 10  |-  -.  (
( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  /\  g : ( ( A  +c  1o )  \  { ( f `
 A ) } ) -1-1-onto-> A )
3534pm2.21i 123 . . . . . . . . 9  |-  ( ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  /\  g : ( ( A  +c  1o )  \  { ( f `
 A ) } ) -1-1-onto-> A )  ->  F.  )
3635ex 423 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  ->  ( g : ( ( A  +c  1o )  \  { ( f `  A ) } ) -1-1-onto-> A  ->  F.  )
)
3736exlimdv 1666 . . . . . . 7  |-  ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  ->  ( E. g 
g : ( ( A  +c  1o ) 
\  { ( f `
 A ) } ) -1-1-onto-> A  ->  F.  )
)
3822, 37mpd 14 . . . . . 6  |-  ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  ->  F.  )
3938ex 423 . . . . 5  |-  ( 1o 
~<  A  ->  ( f : ~P A -1-1-onto-> ( A  +c  1o )  ->  F.  ) )
4039exlimdv 1666 . . . 4  |-  ( 1o 
~<  A  ->  ( E. f  f : ~P A
-1-1-onto-> ( A  +c  1o )  ->  F.  ) )
4111, 40syl5 28 . . 3  |-  ( 1o 
~<  A  ->  ( ( A  +c  1o ) 
~~  ~P A  ->  F.  ) )
428, 41mtoi 169 . 2  |-  ( 1o 
~<  A  ->  -.  ( A  +c  1o )  ~~  ~P A )
43 brsdom 6886 . 2  |-  ( ( A  +c  1o ) 
~<  ~P A  <->  ( ( A  +c  1o )  ~<_  ~P A  /\  -.  ( A  +c  1o )  ~~  ~P A ) )
447, 42, 43sylanbrc 645 1  |-  ( 1o 
~<  A  ->  ( A  +c  1o )  ~<  ~P A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    F. wfal 1308   E.wex 1530    = wceq 1625    e. wcel 1686   A.wral 2545   _Vcvv 2790    \ cdif 3151    C_ wss 3154   (/)c0 3457   ifcif 3567   ~Pcpw 3627   {csn 3642   U.cuni 3829   class class class wbr 4025   {copab 4078    e. cmpt 4079    We wwe 4353    X. cxp 4689   `'ccnv 4690   dom cdm 4691   "cima 4694    o. ccom 4695   -->wf 5253   -1-1-onto->wf1o 5256   ` cfv 5257  (class class class)co 5860   1oc1o 6474   2oc2o 6475    ~~ cen 6862    ~<_ cdom 6863    ~< csdm 6864    +c ccda 7795
This theorem is referenced by:  finngch  8279  gchcda1  8280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-oi 7227  df-card 7574  df-cda 7796
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