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Theorem dfac8b 8443
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 8365 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
2 bren 7562 . . 3  |-  ( (
card `  A )  ~~  A  <->  E. f  f : ( card `  A
)
-1-1-onto-> A )
31, 2sylib 196 . 2  |-  ( A  e.  dom  card  ->  E. f  f : (
card `  A ) -1-1-onto-> A
)
4 sqxpexg 6586 . . . . 5  |-  ( A  e.  dom  card  ->  ( A  X.  A )  e.  _V )
5 incom 3631 . . . . . 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 ) } )
6 inex1g 4536 . . . . . 6  |-  ( ( A  X.  A )  e.  _V  ->  (
( A  X.  A
)  i^i  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) } )  e.  _V )
75, 6syl5eqel 2494 . . . . 5  |-  ( ( A  X.  A )  e.  _V  ->  ( { <. z ,  w >.  |  ( `' f `
 z )  _E  ( `' f `  w ) }  i^i  ( A  X.  A
) )  e.  _V )
84, 7syl 17 . . . 4  |-  ( A  e.  dom  card  ->  ( { <. z ,  w >.  |  ( `' f `
 z )  _E  ( `' f `  w ) }  i^i  ( A  X.  A
) )  e.  _V )
9 f1ocnv 5810 . . . . . 6  |-  ( f : ( card `  A
)
-1-1-onto-> A  ->  `' f : A -1-1-onto-> ( card `  A
) )
10 cardon 8356 . . . . . . . 8  |-  ( card `  A )  e.  On
1110onordi 5513 . . . . . . 7  |-  Ord  ( card `  A )
12 ordwe 5422 . . . . . . 7  |-  ( Ord  ( card `  A
)  ->  _E  We  ( card `  A )
)
1311, 12ax-mp 5 . . . . . 6  |-  _E  We  ( card `  A )
14 eqid 2402 . . . . . . 7  |-  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  =  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }
1514f1owe 6231 . . . . . 6  |-  ( `' f : A -1-1-onto-> ( card `  A )  ->  (  _E  We  ( card `  A
)  ->  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  We  A ) )
169, 13, 15mpisyl 19 . . . . 5  |-  ( f : ( card `  A
)
-1-1-onto-> A  ->  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  We  A )
17 weinxp 4890 . . . . 5  |-  ( {
<. z ,  w >.  |  ( `' f `  z )  _E  ( `' f `  w
) }  We  A  <->  ( { <. z ,  w >.  |  ( `' f `
 z )  _E  ( `' f `  w ) }  i^i  ( A  X.  A
) )  We  A
)
1816, 17sylib 196 . . . 4  |-  ( f : ( card `  A
)
-1-1-onto-> A  ->  ( { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  i^i  ( A  X.  A ) )  We  A )
19 weeq1 4810 . . . . 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 ) )
2019spcegv 3144 . . . 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 )
)
218, 18, 20syl2im 36 . . 3  |-  ( A  e.  dom  card  ->  ( f : ( card `  A ) -1-1-onto-> A  ->  E. x  x  We  A )
)
2221exlimdv 1745 . 2  |-  ( A  e.  dom  card  ->  ( E. f  f : ( card `  A
)
-1-1-onto-> A  ->  E. x  x  We  A ) )
233, 22mpd 15 1  |-  ( A  e.  dom  card  ->  E. x  x  We  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   E.wex 1633    e. wcel 1842   _Vcvv 3058    i^i cin 3412   class class class wbr 4394   {copab 4451    _E cep 4731    We wwe 4780    X. cxp 4820   `'ccnv 4821   dom cdm 4822   Ord word 5408   -1-1-onto->wf1o 5567   ` cfv 5568    ~~ cen 7550   cardccrd 8347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-ord 5412  df-on 5413  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-en 7554  df-card 8351
This theorem is referenced by:  ween  8447  ac5num  8448  dfac8  8546
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