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Theorem canthp1 8292
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 7077 . . . 4  |-  1o  ~<  2o
2 sdomdom 6905 . . . 4  |-  ( 1o 
~<  2o  ->  1o  ~<_  2o )
3 cdadom2 7829 . . . 4  |-  ( 1o  ~<_  2o  ->  ( A  +c  1o )  ~<_  ( A  +c  2o ) )
41, 2, 3mp2b 9 . . 3  |-  ( A  +c  1o )  ~<_  ( A  +c  2o )
5 canthp1lem1 8290 . . 3  |-  ( 1o 
~<  A  ->  ( A  +c  2o )  ~<_  ~P A )
6 domtr 6930 . . 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 6926 . . . . 5  |-  ( ( A  +c  1o ) 
~~  ~P A  ->  ~P A  ~~  ( A  +c  1o ) )
10 bren 6887 . . . . 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 5488 . . . . . . . . . 10  |-  ( f : ~P A -1-1-onto-> ( A  +c  1o )  -> 
f : ~P A --> ( A  +c  1o ) )
13 relsdom 6886 . . . . . . . . . . . 12  |-  Rel  ~<
1413brrelex2i 4746 . . . . . . . . . . 11  |-  ( 1o 
~<  A  ->  A  e. 
_V )
15 pwidg 3650 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  A  e.  ~P A )
1614, 15syl 15 . . . . . . . . . 10  |-  ( 1o 
~<  A  ->  A  e. 
~P A )
17 ffvelrn 5679 . . . . . . . . . 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 7818 . . . . . . . . 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 6887 . . . . . . . 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 2302 . . . . . . . . . . . . . 14  |-  ( w  =  x  ->  (
w  =  A  <->  x  =  A ) )
27 id 19 . . . . . . . . . . . . . 14  |-  ( w  =  x  ->  w  =  x )
2826, 27ifbieq2d 3598 . . . . . . . . . . . . 13  |-  ( w  =  x  ->  if ( w  =  A ,  (/) ,  w )  =  if ( x  =  A ,  (/) ,  x ) )
2928cbvmptv 4127 . . . . . . . . . . . 12  |-  ( w  e.  ~P A  |->  if ( w  =  A ,  (/) ,  w ) )  =  ( x  e.  ~P A  |->  if ( x  =  A ,  (/) ,  x ) )
3029coeq2i 4860 . . . . . . . . . . 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 2296 . . . . . . . . . . . 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 8282 . . . . . . . . . . 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 2296 . . . . . . . . . . 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 8291 . . . . . . . . . 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 1626 . . . . . . 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 1626 . . . 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 6900 . 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 1531    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    \ cdif 3162    C_ wss 3165   (/)c0 3468   ifcif 3578   ~Pcpw 3638   {csn 3653   U.cuni 3843   class class class wbr 4039   {copab 4092    e. cmpt 4093    We wwe 4367    X. cxp 4703   `'ccnv 4704   dom cdm 4705   "cima 4708    o. ccom 4709   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   1oc1o 6488   2oc2o 6489    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878    +c ccda 7809
This theorem is referenced by:  finngch  8293  gchcda1  8294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-cda 7810
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