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Theorem canthnum 8204
Description: The set of well-orderable subsets of a set  A strictly dominates  A. A stronger form of canth2 6947. 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 4132 . . . 4  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 inex1g 4097 . . . 4  |-  ( ~P A  e.  _V  ->  ( ~P A  i^i  Fin )  e.  _V )
3 infpwfidom 7588 . . . 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 4097 . . . . 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 7514 . . . . . 6  |-  ( x  e.  Fin  ->  x  e.  dom  card )
87ssriv 3126 . . . . 5  |-  Fin  C_  dom  card
9 sslin 3337 . . . . 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 6840 . . . 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 6847 . . 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 8199 . . . . . 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 8203 . . . . 5  |-  ( A  e.  V  ->  -.  f : ( ~P A  i^i  dom  card ) -1-1-> A )
20 f1of1 5374 . . . . 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 6843 . . . 4  |-  ( A 
~~  ( ~P A  i^i  dom  card )  ->  ( ~P A  i^i  dom  card )  ~~  A )
24 bren 6804 . . . 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 6817 . 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 2740    i^i cin 3093    C_ wss 3094   ~Pcpw 3566   {csn 3581   U.cuni 3768   class class class wbr 3963   {copab 4016    We wwe 4288    X. cxp 4624   `'ccnv 4625   dom cdm 4626   "cima 4629   -1-1->wf1 4635   -1-1-onto->wf1o 4637   ` cfv 4638    ~~ cen 6793    ~<_ cdom 6794    ~< csdm 6795   Fincfn 6796   cardccrd 7501
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 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-1st 6021  df-iota 6190  df-riota 6237  df-recs 6321  df-1o 6412  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-oi 7158  df-card 7505
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