MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  canthnum Unicode version

Theorem canthnum 8266
Description: The set of well-orderable subsets of a set  A strictly dominates  A. A stronger form of canth2 7009. 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 4193 . . . 4  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 inex1g 4158 . . . 4  |-  ( ~P A  e.  _V  ->  ( ~P A  i^i  Fin )  e.  _V )
3 infpwfidom 7650 . . . 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 4158 . . . . 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 7576 . . . . . 6  |-  ( x  e.  Fin  ->  x  e.  dom  card )
87ssriv 3185 . . . . 5  |-  Fin  C_  dom  card
9 sslin 3396 . . . . 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 6902 . . . 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 6909 . . 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 2284 . . . . . . 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 8261 . . . . . 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 2284 . . . . . 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 2284 . . . . . 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 8265 . . . . 5  |-  ( A  e.  V  ->  -.  f : ( ~P A  i^i  dom  card ) -1-1-> A )
20 f1of1 5436 . . . . 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 1868 . . 3  |-  ( A  e.  V  ->  -.  E. f  f : ( ~P A  i^i  dom  card ) -1-1-onto-> A )
23 ensym 6905 . . . 4  |-  ( A 
~~  ( ~P A  i^i  dom  card )  ->  ( ~P A  i^i  dom  card )  ~~  A )
24 bren 6866 . . . 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 6879 . 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 1533    = wceq 1628    e. wcel 1688   A.wral 2544   _Vcvv 2789    i^i cin 3152    C_ wss 3153   ~Pcpw 3626   {csn 3641   U.cuni 3828   class class class wbr 4024   {copab 4077    We wwe 4350    X. cxp 4686   `'ccnv 4687   dom cdm 4688   "cima 4691   -1-1->wf1 5218   -1-1-onto->wf1o 5220   ` cfv 5221    ~~ cen 6855    ~<_ cdom 6856    ~< csdm 6857   Fincfn 6858   cardccrd 7563
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-1st 6083  df-iota 6252  df-riota 6299  df-recs 6383  df-1o 6474  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-oi 7220  df-card 7567
  Copyright terms: Public domain W3C validator