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Theorem canthp1 8485
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 7266 . . . 4  |-  1o  ~<  2o
2 sdomdom 7094 . . . 4  |-  ( 1o 
~<  2o  ->  1o  ~<_  2o )
3 cdadom2 8023 . . . 4  |-  ( 1o  ~<_  2o  ->  ( A  +c  1o )  ~<_  ( A  +c  2o ) )
41, 2, 3mp2b 10 . . 3  |-  ( A  +c  1o )  ~<_  ( A  +c  2o )
5 canthp1lem1 8483 . . 3  |-  ( 1o 
~<  A  ->  ( A  +c  2o )  ~<_  ~P A )
6 domtr 7119 . . 3  |-  ( ( ( A  +c  1o )  ~<_  ( A  +c  2o )  /\  ( A  +c  2o )  ~<_  ~P A )  ->  ( A  +c  1o )  ~<_  ~P A )
74, 5, 6sylancr 645 . 2  |-  ( 1o 
~<  A  ->  ( A  +c  1o )  ~<_  ~P A )
8 fal 1328 . . 3  |-  -.  F.
9 ensym 7115 . . . . 5  |-  ( ( A  +c  1o ) 
~~  ~P A  ->  ~P A  ~~  ( A  +c  1o ) )
10 bren 7076 . . . . 5  |-  ( ~P A  ~~  ( A  +c  1o )  <->  E. f 
f : ~P A -1-1-onto-> ( A  +c  1o ) )
119, 10sylib 189 . . . 4  |-  ( ( A  +c  1o ) 
~~  ~P A  ->  E. f 
f : ~P A -1-1-onto-> ( A  +c  1o ) )
12 f1of 5633 . . . . . . . . . 10  |-  ( f : ~P A -1-1-onto-> ( A  +c  1o )  -> 
f : ~P A --> ( A  +c  1o ) )
13 relsdom 7075 . . . . . . . . . . . 12  |-  Rel  ~<
1413brrelex2i 4878 . . . . . . . . . . 11  |-  ( 1o 
~<  A  ->  A  e. 
_V )
15 pwidg 3771 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  A  e.  ~P A )
1614, 15syl 16 . . . . . . . . . 10  |-  ( 1o 
~<  A  ->  A  e. 
~P A )
17 ffvelrn 5827 . . . . . . . . . 10  |-  ( ( f : ~P A --> ( A  +c  1o )  /\  A  e.  ~P A )  ->  (
f `  A )  e.  ( A  +c  1o ) )
1812, 16, 17syl2anr 465 . . . . . . . . 9  |-  ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  ->  ( f `  A )  e.  ( A  +c  1o ) )
19 cda1dif 8012 . . . . . . . . 9  |-  ( ( f `  A )  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { ( f `
 A ) } )  ~~  A )
2018, 19syl 16 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  ->  ( ( A  +c  1o )  \  { ( f `  A ) } ) 
~~  A )
21 bren 7076 . . . . . . . 8  |-  ( ( ( A  +c  1o )  \  { ( f `
 A ) } )  ~~  A  <->  E. g 
g : ( ( A  +c  1o ) 
\  { ( f `
 A ) } ) -1-1-onto-> A )
2220, 21sylib 189 . . . . . . 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 731 . . . . . . . . 9  |-  ( ( ( 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 732 . . . . . . . . 9  |-  ( ( ( 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 448 . . . . . . . . 9  |-  ( ( ( 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 2410 . . . . . . . . . . . 12  |-  ( w  =  x  ->  (
w  =  A  <->  x  =  A ) )
27 id 20 . . . . . . . . . . . 12  |-  ( w  =  x  ->  w  =  x )
2826, 27ifbieq2d 3719 . . . . . . . . . . 11  |-  ( w  =  x  ->  if ( w  =  A ,  (/) ,  w )  =  if ( x  =  A ,  (/) ,  x ) )
2928cbvmptv 4260 . . . . . . . . . 10  |-  ( w  e.  ~P A  |->  if ( w  =  A ,  (/) ,  w ) )  =  ( x  e.  ~P A  |->  if ( x  =  A ,  (/) ,  x ) )
3029coeq2i 4992 . . . . . . . . 9  |-  ( ( 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 2404 . . . . . . . . . 10  |-  { <. 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 8475 . . . . . . . . 9  |-  { <. 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 2404 . . . . . . . . 9  |-  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 8484 . . . . . . . 8  |-  -.  (
( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  /\  g : ( ( A  +c  1o )  \  { ( f `
 A ) } ) -1-1-onto-> A )
3534pm2.21i 125 . . . . . . 7  |-  ( ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  /\  g : ( ( A  +c  1o )  \  { ( f `
 A ) } ) -1-1-onto-> A )  ->  F.  )
3622, 35exlimddv 1645 . . . . . 6  |-  ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  ->  F.  )
3736ex 424 . . . . 5  |-  ( 1o 
~<  A  ->  ( f : ~P A -1-1-onto-> ( A  +c  1o )  ->  F.  ) )
3837exlimdv 1643 . . . 4  |-  ( 1o 
~<  A  ->  ( E. f  f : ~P A
-1-1-onto-> ( A  +c  1o )  ->  F.  ) )
3911, 38syl5 30 . . 3  |-  ( 1o 
~<  A  ->  ( ( A  +c  1o ) 
~~  ~P A  ->  F.  ) )
408, 39mtoi 171 . 2  |-  ( 1o 
~<  A  ->  -.  ( A  +c  1o )  ~~  ~P A )
41 brsdom 7089 . 2  |-  ( ( A  +c  1o ) 
~<  ~P A  <->  ( ( A  +c  1o )  ~<_  ~P A  /\  -.  ( A  +c  1o )  ~~  ~P A ) )
427, 40, 41sylanbrc 646 1  |-  ( 1o 
~<  A  ->  ( A  +c  1o )  ~<  ~P A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    F. wfal 1323   E.wex 1547    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    \ cdif 3277    C_ wss 3280   (/)c0 3588   ifcif 3699   ~Pcpw 3759   {csn 3774   U.cuni 3975   class class class wbr 4172   {copab 4225    e. cmpt 4226    We wwe 4500    X. cxp 4835   `'ccnv 4836   dom cdm 4837   "cima 4840    o. ccom 4841   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   1oc1o 6676   2oc2o 6677    ~~ cen 7065    ~<_ cdom 7066    ~< csdm 7067    +c ccda 8003
This theorem is referenced by:  finngch  8486  gchcda1  8487
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
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  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-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-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-card 7782  df-cda 8004
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