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Theorem dfac8b 7904
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 7832 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
2 bren 7109 . . 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 4981 . . . . . 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 3525 . . . . . 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 4338 . . . . . 6  |-  ( ( A  X.  A )  e.  _V  ->  (
( A  X.  A
)  i^i  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) } )  e.  _V )
86, 7syl5eqel 2519 . . . . 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 5679 . . . . . 6  |-  ( f : ( card `  A
)
-1-1-onto-> A  ->  `' f : A -1-1-onto-> ( card `  A
) )
11 cardon 7823 . . . . . . . 8  |-  ( card `  A )  e.  On
1211onordi 4678 . . . . . . 7  |-  Ord  ( card `  A )
13 ordwe 4586 . . . . . . 7  |-  ( Ord  ( card `  A
)  ->  _E  We  ( card `  A )
)
1412, 13ax-mp 8 . . . . . 6  |-  _E  We  ( card `  A )
15 eqid 2435 . . . . . . 7  |-  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  =  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }
1615f1owe 6065 . . . . . 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 4937 . . . . 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 4562 . . . . 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 3029 . . . 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 2948    i^i cin 3311   class class class wbr 4204   {copab 4257    _E cep 4484    We wwe 4532   Ord word 4572    X. cxp 4868   `'ccnv 4869   dom cdm 4870   -1-1-onto->wf1o 5445   ` cfv 5446    ~~ cen 7098   cardccrd 7814
This theorem is referenced by:  ween  7908  ac5num  7909  dfac8  8007
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 4693
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-rab 2706  df-v 2950  df-sbc 3154  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-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-we 4535  df-ord 4576  df-on 4577  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-en 7102  df-card 7818
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