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Theorem canthp1 8156
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
StepHypRef Expression
1 1sdom2 6946 . . . 4  |-  1o  ~<  2o
2 sdomdom 6775 . . . 4  |-  ( 1o 
~<  2o  ->  1o  ~<_  2o )
3 cdadom2 7697 . . . 4  |-  ( 1o  ~<_  2o  ->  ( A  +c  1o )  ~<_  ( A  +c  2o ) )
41, 2, 3mp2b 11 . . 3  |-  ( A  +c  1o )  ~<_  ( A  +c  2o )
5 canthp1lem1 8154 . . 3  |-  ( 1o 
~<  A  ->  ( A  +c  2o )  ~<_  ~P A )
6 domtr 6799 . . 3  |-  ( ( ( A  +c  1o )  ~<_  ( A  +c  2o )  /\  ( A  +c  2o )  ~<_  ~P A )  ->  ( A  +c  1o )  ~<_  ~P A )
74, 5, 6sylancr 647 . 2  |-  ( 1o 
~<  A  ->  ( A  +c  1o )  ~<_  ~P A )
8 fal 1319 . . 3  |-  -.  F.
9 ensym 6796 . . . . 5  |-  ( ( A  +c  1o ) 
~~  ~P A  ->  ~P A  ~~  ( A  +c  1o ) )
10 bren 6757 . . . . 5  |-  ( ~P A  ~~  ( A  +c  1o )  <->  E. f 
f : ~P A -1-1-onto-> ( A  +c  1o ) )
119, 10sylib 190 . . . 4  |-  ( ( A  +c  1o ) 
~~  ~P A  ->  E. f 
f : ~P A -1-1-onto-> ( A  +c  1o ) )
12 f1of 5329 . . . . . . . . . 10  |-  ( f : ~P A -1-1-onto-> ( A  +c  1o )  -> 
f : ~P A --> ( A  +c  1o ) )
13 relsdom 6756 . . . . . . . . . . . 12  |-  Rel  ~<
1413brrelex2i 4637 . . . . . . . . . . 11  |-  ( 1o 
~<  A  ->  A  e. 
_V )
15 pwidg 3541 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  A  e.  ~P A )
1614, 15syl 17 . . . . . . . . . 10  |-  ( 1o 
~<  A  ->  A  e. 
~P A )
17 ffvelrn 5515 . . . . . . . . . 10  |-  ( ( f : ~P A --> ( A  +c  1o )  /\  A  e.  ~P A )  ->  (
f `  A )  e.  ( A  +c  1o ) )
1812, 16, 17syl2anr 466 . . . . . . . . 9  |-  ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  ->  ( f `  A )  e.  ( A  +c  1o ) )
19 cda1dif 7686 . . . . . . . . 9  |-  ( ( f `  A )  e.  ( A  +c  1o )  ->  ( ( A  +c  1o ) 
\  { ( f `
 A ) } )  ~~  A )
2018, 19syl 17 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  ->  ( ( A  +c  1o )  \  { ( f `  A ) } ) 
~~  A )
21 bren 6757 . . . . . . . 8  |-  ( ( ( A  +c  1o )  \  { ( f `
 A ) } )  ~~  A  <->  E. g 
g : ( ( A  +c  1o ) 
\  { ( f `
 A ) } ) -1-1-onto-> A )
2220, 21sylib 190 . . . . . . 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 733 . . . . . . . . . . 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 734 . . . . . . . . . . 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 449 . . . . . . . . . . 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 2259 . . . . . . . . . . . . . 14  |-  ( w  =  x  ->  (
w  =  A  <->  x  =  A ) )
27 id 21 . . . . . . . . . . . . . 14  |-  ( w  =  x  ->  w  =  x )
2826, 27ifbieq2d 3490 . . . . . . . . . . . . 13  |-  ( w  =  x  ->  if ( w  =  A ,  (/) ,  w )  =  if ( x  =  A ,  (/) ,  x ) )
2928cbvmptv 4008 . . . . . . . . . . . 12  |-  ( w  e.  ~P A  |->  if ( w  =  A ,  (/) ,  w ) )  =  ( x  e.  ~P A  |->  if ( x  =  A ,  (/) ,  x ) )
3029coeq2i 4751 . . . . . . . . . . 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 2253 . . . . . . . . . . . 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 8146 . . . . . . . . . . 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 2253 . . . . . . . . . . 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 8155 . . . . . . . . . 10  |-  -.  (
( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  /\  g : ( ( A  +c  1o )  \  { ( f `
 A ) } ) -1-1-onto-> A )
3534pm2.21i 125 . . . . . . . . 9  |-  ( ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  /\  g : ( ( A  +c  1o )  \  { ( f `
 A ) } ) -1-1-onto-> A )  ->  F.  )
3635ex 425 . . . . . . . 8  |-  ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  ->  ( g : ( ( A  +c  1o )  \  { ( f `  A ) } ) -1-1-onto-> A  ->  F.  )
)
3736exlimdv 1932 . . . . . . 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 16 . . . . . 6  |-  ( ( 1o  ~<  A  /\  f : ~P A -1-1-onto-> ( A  +c  1o ) )  ->  F.  )
3938ex 425 . . . . 5  |-  ( 1o 
~<  A  ->  ( f : ~P A -1-1-onto-> ( A  +c  1o )  ->  F.  ) )
4039exlimdv 1932 . . . 4  |-  ( 1o 
~<  A  ->  ( E. f  f : ~P A
-1-1-onto-> ( A  +c  1o )  ->  F.  ) )
4111, 40syl5 30 . . 3  |-  ( 1o 
~<  A  ->  ( ( A  +c  1o ) 
~~  ~P A  ->  F.  ) )
428, 41mtoi 171 . 2  |-  ( 1o 
~<  A  ->  -.  ( A  +c  1o )  ~~  ~P A )
43 brsdom 6770 . 2  |-  ( ( A  +c  1o ) 
~<  ~P A  <->  ( ( A  +c  1o )  ~<_  ~P A  /\  -.  ( A  +c  1o )  ~~  ~P A ) )
447, 42, 43sylanbrc 648 1  |-  ( 1o 
~<  A  ->  ( A  +c  1o )  ~<  ~P A )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    F. wfal 1313   E.wex 1537    = wceq 1619    e. wcel 1621   A.wral 2509   _Vcvv 2727    \ cdif 3075    C_ wss 3078   (/)c0 3362   ifcif 3470   ~Pcpw 3530   {csn 3544   U.cuni 3727   class class class wbr 3920   {copab 3973    e. cmpt 3974    We wwe 4244    X. cxp 4578   `'ccnv 4579   dom cdm 4580   "cima 4583    o. ccom 4584   -->wf 4588   -1-1-onto->wf1o 4591   ` cfv 4592  (class class class)co 5710   1oc1o 6358   2oc2o 6359    ~~ cen 6746    ~<_ cdom 6747    ~< csdm 6748    +c ccda 7677
This theorem is referenced by:  finngch  8157  gchcda1  8158
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-2o 6366  df-oadd 6369  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-oi 7109  df-card 7456  df-cda 7678
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