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Theorem canthnum 8225
Description: The set of well-orderable subsets of a set  A strictly dominates  A. A stronger form of canth2 6968. Corollary 1.4(a) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 19-May-2015.)
Assertion
Ref Expression
canthnum  |-  ( A  e.  V  ->  A  ~<  ( ~P A  i^i  dom 
card ) )

Proof of Theorem canthnum
StepHypRef Expression
1 pwexg 4152 . . . 4  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 inex1g 4117 . . . 4  |-  ( ~P A  e.  _V  ->  ( ~P A  i^i  Fin )  e.  _V )
3 infpwfidom 7609 . . . 4  |-  ( ( ~P A  i^i  Fin )  e.  _V  ->  A  ~<_  ( ~P A  i^i  Fin ) )
41, 2, 33syl 20 . . 3  |-  ( A  e.  V  ->  A  ~<_  ( ~P A  i^i  Fin ) )
5 inex1g 4117 . . . . 5  |-  ( ~P A  e.  _V  ->  ( ~P A  i^i  dom  card )  e.  _V )
61, 5syl 17 . . . 4  |-  ( A  e.  V  ->  ( ~P A  i^i  dom  card )  e.  _V )
7 finnum 7535 . . . . . 6  |-  ( x  e.  Fin  ->  x  e.  dom  card )
87ssriv 3145 . . . . 5  |-  Fin  C_  dom  card
9 sslin 3356 . . . . 5  |-  ( Fin  C_  dom  card  ->  ( ~P A  i^i  Fin )  C_  ( ~P A  i^i  dom 
card ) )
108, 9ax-mp 10 . . . 4  |-  ( ~P A  i^i  Fin )  C_  ( ~P A  i^i  dom 
card )
11 ssdomg 6861 . . . 4  |-  ( ( ~P A  i^i  dom  card )  e.  _V  ->  ( ( ~P A  i^i  Fin )  C_  ( ~P A  i^i  dom  card )  -> 
( ~P A  i^i  Fin )  ~<_  ( ~P A  i^i  dom  card ) ) )
126, 10, 11ee10 1372 . . 3  |-  ( A  e.  V  ->  ( ~P A  i^i  Fin )  ~<_  ( ~P A  i^i  dom  card ) )
13 domtr 6868 . . 3  |-  ( ( A  ~<_  ( ~P A  i^i  Fin )  /\  ( ~P A  i^i  Fin )  ~<_  ( ~P A  i^i  dom  card ) )  ->  A  ~<_  ( ~P A  i^i  dom  card ) )
144, 12, 13syl2anc 645 . 2  |-  ( A  e.  V  ->  A  ~<_  ( ~P A  i^i  dom  card ) )
15 eqid 2256 . . . . . . 7  |-  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( f `  ( `' r " { y } ) )  =  y ) ) }  =  { <. x ,  r >.  |  ( ( x 
C_  A  /\  r  C_  ( x  X.  x
) )  /\  (
r  We  x  /\  A. y  e.  x  ( f `  ( `' r " { y } ) )  =  y ) ) }
1615fpwwecbv 8220 . . . . . 6  |-  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( f `  ( `' r " { y } ) )  =  y ) ) }  =  { <. a ,  s >.  |  ( ( a 
C_  A  /\  s  C_  ( a  X.  a
) )  /\  (
s  We  a  /\  A. z  e.  a  ( f `  ( `' s " { z } ) )  =  z ) ) }
17 eqid 2256 . . . . . 6  |-  U. dom  {
<. x ,  r >.  |  ( ( x 
C_  A  /\  r  C_  ( x  X.  x
) )  /\  (
r  We  x  /\  A. y  e.  x  ( f `  ( `' r " { y } ) )  =  y ) ) }  =  U. dom  { <. x ,  r >.  |  ( ( x 
C_  A  /\  r  C_  ( x  X.  x
) )  /\  (
r  We  x  /\  A. y  e.  x  ( f `  ( `' r " { y } ) )  =  y ) ) }
18 eqid 2256 . . . . . 6  |-  ( `' ( { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( f `  ( `' r " { y } ) )  =  y ) ) } `  U. dom  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( f `  ( `' r " {
y } ) )  =  y ) ) } ) " {
( f `  U. dom  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( f `  ( `' r " {
y } ) )  =  y ) ) } ) } )  =  ( `' ( { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( f `  ( `' r " {
y } ) )  =  y ) ) } `  U. dom  {
<. x ,  r >.  |  ( ( x 
C_  A  /\  r  C_  ( x  X.  x
) )  /\  (
r  We  x  /\  A. y  e.  x  ( f `  ( `' r " { y } ) )  =  y ) ) } ) " { ( f `  U. dom  {
<. x ,  r >.  |  ( ( x 
C_  A  /\  r  C_  ( x  X.  x
) )  /\  (
r  We  x  /\  A. y  e.  x  ( f `  ( `' r " { y } ) )  =  y ) ) } ) } )
1916, 17, 18canthnumlem 8224 . . . . 5  |-  ( A  e.  V  ->  -.  f : ( ~P A  i^i  dom  card ) -1-1-> A )
20 f1of1 5395 . . . . 5  |-  ( f : ( ~P A  i^i  dom  card ) -1-1-onto-> A  ->  f :
( ~P A  i^i  dom 
card ) -1-1-> A )
2119, 20nsyl 115 . . . 4  |-  ( A  e.  V  ->  -.  f : ( ~P A  i^i  dom  card ) -1-1-onto-> A )
2221nexdv 2061 . . 3  |-  ( A  e.  V  ->  -.  E. f  f : ( ~P A  i^i  dom  card ) -1-1-onto-> A )
23 ensym 6864 . . . 4  |-  ( A 
~~  ( ~P A  i^i  dom  card )  ->  ( ~P A  i^i  dom  card )  ~~  A )
24 bren 6825 . . . 4  |-  ( ( ~P A  i^i  dom  card )  ~~  A  <->  E. f 
f : ( ~P A  i^i  dom  card )
-1-1-onto-> A )
2523, 24sylib 190 . . 3  |-  ( A 
~~  ( ~P A  i^i  dom  card )  ->  E. f 
f : ( ~P A  i^i  dom  card )
-1-1-onto-> A )
2622, 25nsyl 115 . 2  |-  ( A  e.  V  ->  -.  A  ~~  ( ~P A  i^i  dom  card ) )
27 brsdom 6838 . 2  |-  ( A 
~<  ( ~P A  i^i  dom 
card )  <->  ( A  ~<_  ( ~P A  i^i  dom  card )  /\  -.  A  ~~  ( ~P A  i^i  dom 
card ) ) )
2814, 26, 27sylanbrc 648 1  |-  ( A  e.  V  ->  A  ~<  ( ~P A  i^i  dom 
card ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   A.wral 2516   _Vcvv 2757    i^i cin 3112    C_ wss 3113   ~Pcpw 3585   {csn 3600   U.cuni 3787   class class class wbr 3983   {copab 4036    We wwe 4309    X. cxp 4645   `'ccnv 4646   dom cdm 4647   "cima 4650   -1-1->wf1 4656   -1-1-onto->wf1o 4658   ` cfv 4659    ~~ cen 6814    ~<_ cdom 6815    ~< csdm 6816   Fincfn 6817   cardccrd 7522
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 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-1st 6042  df-iota 6211  df-riota 6258  df-recs 6342  df-1o 6433  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-oi 7179  df-card 7526
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