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Theorem dfac8b 7658
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 7586 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
2 bren 6871 . . 3  |-  ( (
card `  A )  ~~  A  <->  E. f  f : ( card `  A
)
-1-1-onto-> A )
31, 2sylib 188 . 2  |-  ( A  e.  dom  card  ->  E. f  f : (
card `  A ) -1-1-onto-> A
)
4 xpexg 4800 . . . . . 6  |-  ( ( A  e.  dom  card  /\  A  e.  dom  card )  ->  ( A  X.  A )  e.  _V )
54anidms 626 . . . . 5  |-  ( A  e.  dom  card  ->  ( A  X.  A )  e.  _V )
6 incom 3361 . . . . . 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 4157 . . . . . 6  |-  ( ( A  X.  A )  e.  _V  ->  (
( A  X.  A
)  i^i  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) } )  e.  _V )
86, 7syl5eqel 2367 . . . . 5  |-  ( ( A  X.  A )  e.  _V  ->  ( { <. z ,  w >.  |  ( `' f `
 z )  _E  ( `' f `  w ) }  i^i  ( A  X.  A
) )  e.  _V )
95, 8syl 15 . . . 4  |-  ( A  e.  dom  card  ->  ( { <. z ,  w >.  |  ( `' f `
 z )  _E  ( `' f `  w ) }  i^i  ( A  X.  A
) )  e.  _V )
10 f1ocnv 5485 . . . . . 6  |-  ( f : ( card `  A
)
-1-1-onto-> A  ->  `' f : A -1-1-onto-> ( card `  A
) )
11 cardon 7577 . . . . . . . 8  |-  ( card `  A )  e.  On
1211onordi 4497 . . . . . . 7  |-  Ord  ( card `  A )
13 ordwe 4405 . . . . . . 7  |-  ( Ord  ( card `  A
)  ->  _E  We  ( card `  A )
)
1412, 13ax-mp 8 . . . . . 6  |-  _E  We  ( card `  A )
15 eqid 2283 . . . . . . 7  |-  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  =  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }
1615f1owe 5850 . . . . . 6  |-  ( `' f : A -1-1-onto-> ( card `  A )  ->  (  _E  We  ( card `  A
)  ->  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  We  A ) )
1710, 14, 16ee10 1366 . . . . 5  |-  ( f : ( card `  A
)
-1-1-onto-> A  ->  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  We  A )
18 weinxp 4757 . . . . 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 188 . . . 4  |-  ( f : ( card `  A
)
-1-1-onto-> A  ->  ( { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  i^i  ( A  X.  A ) )  We  A )
20 weeq1 4381 . . . . 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 2869 . . . 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 34 . . 3  |-  ( A  e.  dom  card  ->  ( f : ( card `  A ) -1-1-onto-> A  ->  E. x  x  We  A )
)
2322exlimdv 1664 . 2  |-  ( A  e.  dom  card  ->  ( E. f  f : ( card `  A
)
-1-1-onto-> A  ->  E. x  x  We  A ) )
243, 23mpd 14 1  |-  ( A  e.  dom  card  ->  E. x  x  We  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1528    e. wcel 1684   _Vcvv 2788    i^i cin 3151   class class class wbr 4023   {copab 4076    _E cep 4303    We wwe 4351   Ord word 4391    X. cxp 4687   `'ccnv 4688   dom cdm 4689   -1-1-onto->wf1o 5254   ` cfv 5255    ~~ cen 6860   cardccrd 7568
This theorem is referenced by:  ween  7662  ac5num  7663  dfac8  7761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-en 6864  df-card 7572
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