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Theorem dfac8b 7912
Description: The well-ordering theorem: every numerable set is well-orderable. (Contributed by Mario Carneiro, 5-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
dfac8b  |-  ( A  e.  dom  card  ->  E. x  x  We  A
)
Distinct variable group:    x, A

Proof of Theorem dfac8b
Dummy variables  w  f  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cardid2 7840 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
2 bren 7117 . . 3  |-  ( (
card `  A )  ~~  A  <->  E. f  f : ( card `  A
)
-1-1-onto-> A )
31, 2sylib 189 . 2  |-  ( A  e.  dom  card  ->  E. f  f : (
card `  A ) -1-1-onto-> A
)
4 xpexg 4989 . . . . . 6  |-  ( ( A  e.  dom  card  /\  A  e.  dom  card )  ->  ( A  X.  A )  e.  _V )
54anidms 627 . . . . 5  |-  ( A  e.  dom  card  ->  ( A  X.  A )  e.  _V )
6 incom 3533 . . . . . 6  |-  ( {
<. z ,  w >.  |  ( `' f `  z )  _E  ( `' f `  w
) }  i^i  ( A  X.  A ) )  =  ( ( A  X.  A )  i^i 
{ <. z ,  w >.  |  ( `' f `
 z )  _E  ( `' f `  w ) } )
7 inex1g 4346 . . . . . 6  |-  ( ( A  X.  A )  e.  _V  ->  (
( A  X.  A
)  i^i  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) } )  e.  _V )
86, 7syl5eqel 2520 . . . . 5  |-  ( ( A  X.  A )  e.  _V  ->  ( { <. z ,  w >.  |  ( `' f `
 z )  _E  ( `' f `  w ) }  i^i  ( A  X.  A
) )  e.  _V )
95, 8syl 16 . . . 4  |-  ( A  e.  dom  card  ->  ( { <. z ,  w >.  |  ( `' f `
 z )  _E  ( `' f `  w ) }  i^i  ( A  X.  A
) )  e.  _V )
10 f1ocnv 5687 . . . . . 6  |-  ( f : ( card `  A
)
-1-1-onto-> A  ->  `' f : A -1-1-onto-> ( card `  A
) )
11 cardon 7831 . . . . . . . 8  |-  ( card `  A )  e.  On
1211onordi 4686 . . . . . . 7  |-  Ord  ( card `  A )
13 ordwe 4594 . . . . . . 7  |-  ( Ord  ( card `  A
)  ->  _E  We  ( card `  A )
)
1412, 13ax-mp 8 . . . . . 6  |-  _E  We  ( card `  A )
15 eqid 2436 . . . . . . 7  |-  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  =  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }
1615f1owe 6073 . . . . . 6  |-  ( `' f : A -1-1-onto-> ( card `  A )  ->  (  _E  We  ( card `  A
)  ->  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  We  A ) )
1710, 14, 16ee10 1385 . . . . 5  |-  ( f : ( card `  A
)
-1-1-onto-> A  ->  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  We  A )
18 weinxp 4945 . . . . 5  |-  ( {
<. z ,  w >.  |  ( `' f `  z )  _E  ( `' f `  w
) }  We  A  <->  ( { <. z ,  w >.  |  ( `' f `
 z )  _E  ( `' f `  w ) }  i^i  ( A  X.  A
) )  We  A
)
1917, 18sylib 189 . . . 4  |-  ( f : ( card `  A
)
-1-1-onto-> A  ->  ( { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  i^i  ( A  X.  A ) )  We  A )
20 weeq1 4570 . . . . 5  |-  ( x  =  ( { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  i^i  ( A  X.  A ) )  ->  ( x  We  A  <->  ( { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  i^i  ( A  X.  A ) )  We  A ) )
2120spcegv 3037 . . . 4  |-  ( ( { <. z ,  w >.  |  ( `' f `
 z )  _E  ( `' f `  w ) }  i^i  ( A  X.  A
) )  e.  _V  ->  ( ( { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  i^i  ( A  X.  A ) )  We  A  ->  E. x  x  We  A )
)
229, 19, 21syl2im 36 . . 3  |-  ( A  e.  dom  card  ->  ( f : ( card `  A ) -1-1-onto-> A  ->  E. x  x  We  A )
)
2322exlimdv 1646 . 2  |-  ( A  e.  dom  card  ->  ( E. f  f : ( card `  A
)
-1-1-onto-> A  ->  E. x  x  We  A ) )
243, 23mpd 15 1  |-  ( A  e.  dom  card  ->  E. x  x  We  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1550    e. wcel 1725   _Vcvv 2956    i^i cin 3319   class class class wbr 4212   {copab 4265    _E cep 4492    We wwe 4540   Ord word 4580    X. cxp 4876   `'ccnv 4877   dom cdm 4878   -1-1-onto->wf1o 5453   ` cfv 5454    ~~ cen 7106   cardccrd 7822
This theorem is referenced by:  ween  7916  ac5num  7917  dfac8  8015
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-en 7110  df-card 7826
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