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Theorem canthnum 8513
Description: The set of well-orderable subsets of a set  A strictly dominates  A. A stronger form of canth2 7251. 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 4375 . . . 4  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 inex1g 4338 . . . 4  |-  ( ~P A  e.  _V  ->  ( ~P A  i^i  Fin )  e.  _V )
3 infpwfidom 7898 . . . 4  |-  ( ( ~P A  i^i  Fin )  e.  _V  ->  A  ~<_  ( ~P A  i^i  Fin ) )
41, 2, 33syl 19 . . 3  |-  ( A  e.  V  ->  A  ~<_  ( ~P A  i^i  Fin ) )
5 inex1g 4338 . . . . 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 7824 . . . . . 6  |-  ( x  e.  Fin  ->  x  e.  dom  card )
87ssriv 3344 . . . . 5  |-  Fin  C_  dom  card
9 sslin 3559 . . . . 5  |-  ( Fin  C_  dom  card  ->  ( ~P A  i^i  Fin )  C_  ( ~P A  i^i  dom 
card ) )
108, 9ax-mp 8 . . . 4  |-  ( ~P A  i^i  Fin )  C_  ( ~P A  i^i  dom 
card )
11 ssdomg 7144 . . . 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 1385 . . 3  |-  ( A  e.  V  ->  ( ~P A  i^i  Fin )  ~<_  ( ~P A  i^i  dom  card ) )
13 domtr 7151 . . 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 643 . 2  |-  ( A  e.  V  ->  A  ~<_  ( ~P A  i^i  dom  card ) )
15 eqid 2435 . . . . . . 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 8508 . . . . . 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 2435 . . . . . 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 2435 . . . . . 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 8512 . . . . 5  |-  ( A  e.  V  ->  -.  f : ( ~P A  i^i  dom  card ) -1-1-> A )
20 f1of1 5664 . . . . 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 1941 . . 3  |-  ( A  e.  V  ->  -.  E. f  f : ( ~P A  i^i  dom  card ) -1-1-onto-> A )
23 ensym 7147 . . . 4  |-  ( A 
~~  ( ~P A  i^i  dom  card )  ->  ( ~P A  i^i  dom  card )  ~~  A )
24 bren 7108 . . . 4  |-  ( ( ~P A  i^i  dom  card )  ~~  A  <->  E. f 
f : ( ~P A  i^i  dom  card )
-1-1-onto-> A )
2523, 24sylib 189 . . 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 7121 . 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 646 1  |-  ( A  e.  V  ->  A  ~<  ( ~P A  i^i  dom 
card ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   {csn 3806   U.cuni 4007   class class class wbr 4204   {copab 4257    We wwe 4532    X. cxp 4867   `'ccnv 4868   dom cdm 4869   "cima 4872   -1-1->wf1 5442   -1-1-onto->wf1o 5444   ` cfv 5445    ~~ cen 7097    ~<_ cdom 7098    ~< csdm 7099   Fincfn 7100   cardccrd 7811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-1st 6340  df-riota 6540  df-recs 6624  df-1o 6715  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-oi 7468  df-card 7815
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