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Theorem canthnum 9027
Description: The set of well-orderable subsets of a set  A strictly dominates  A. A stronger form of canth2 7670. 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
Dummy variables  f 
a  r  s  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 4631 . . . 4  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 inex1g 4590 . . . 4  |-  ( ~P A  e.  _V  ->  ( ~P A  i^i  Fin )  e.  _V )
3 infpwfidom 8409 . . . 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 4590 . . . . 5  |-  ( ~P A  e.  _V  ->  ( ~P A  i^i  dom  card )  e.  _V )
61, 5syl 16 . . . 4  |-  ( A  e.  V  ->  ( ~P A  i^i  dom  card )  e.  _V )
7 finnum 8329 . . . . . 6  |-  ( x  e.  Fin  ->  x  e.  dom  card )
87ssriv 3508 . . . . 5  |-  Fin  C_  dom  card
9 sslin 3724 . . . . 5  |-  ( Fin  C_  dom  card  ->  ( ~P A  i^i  Fin )  C_  ( ~P A  i^i  dom 
card ) )
108, 9ax-mp 5 . . . 4  |-  ( ~P A  i^i  Fin )  C_  ( ~P A  i^i  dom 
card )
11 ssdomg 7561 . . . 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, 11mpisyl 18 . . 3  |-  ( A  e.  V  ->  ( ~P A  i^i  Fin )  ~<_  ( ~P A  i^i  dom  card ) )
13 domtr 7568 . . 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 661 . 2  |-  ( A  e.  V  ->  A  ~<_  ( ~P A  i^i  dom  card ) )
15 eqid 2467 . . . . . . 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 9022 . . . . . 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 2467 . . . . . 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 2467 . . . . . 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 9026 . . . . 5  |-  ( A  e.  V  ->  -.  f : ( ~P A  i^i  dom  card ) -1-1-> A )
20 f1of1 5815 . . . . 5  |-  ( f : ( ~P A  i^i  dom  card ) -1-1-onto-> A  ->  f :
( ~P A  i^i  dom 
card ) -1-1-> A )
2119, 20nsyl 121 . . . 4  |-  ( A  e.  V  ->  -.  f : ( ~P A  i^i  dom  card ) -1-1-onto-> A )
2221nexdv 1832 . . 3  |-  ( A  e.  V  ->  -.  E. f  f : ( ~P A  i^i  dom  card ) -1-1-onto-> A )
23 ensym 7564 . . . 4  |-  ( A 
~~  ( ~P A  i^i  dom  card )  ->  ( ~P A  i^i  dom  card )  ~~  A )
24 bren 7525 . . . 4  |-  ( ( ~P A  i^i  dom  card )  ~~  A  <->  E. f 
f : ( ~P A  i^i  dom  card )
-1-1-onto-> A )
2523, 24sylib 196 . . 3  |-  ( A 
~~  ( ~P A  i^i  dom  card )  ->  E. f 
f : ( ~P A  i^i  dom  card )
-1-1-onto-> A )
2622, 25nsyl 121 . 2  |-  ( A  e.  V  ->  -.  A  ~~  ( ~P A  i^i  dom  card ) )
27 brsdom 7538 . 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 664 1  |-  ( A  e.  V  ->  A  ~<  ( ~P A  i^i  dom 
card ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   A.wral 2814   _Vcvv 3113    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   {csn 4027   U.cuni 4245   class class class wbr 4447   {copab 4504    We wwe 4837    X. cxp 4997   `'ccnv 4998   dom cdm 4999   "cima 5002   -1-1->wf1 5585   -1-1-onto->wf1o 5587   ` cfv 5588    ~~ cen 7513    ~<_ cdom 7514    ~< csdm 7515   Fincfn 7516   cardccrd 8316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-om 6685  df-1st 6784  df-recs 7042  df-1o 7130  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-oi 7935  df-card 8320
This theorem is referenced by: (None)
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