MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfac8b Unicode version

Theorem dfac8b 7838
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 7766 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
2 bren 7046 . . 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 4922 . . . . . 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 3469 . . . . . 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 4280 . . . . . 6  |-  ( ( A  X.  A )  e.  _V  ->  (
( A  X.  A
)  i^i  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) } )  e.  _V )
86, 7syl5eqel 2464 . . . . 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 5620 . . . . . 6  |-  ( f : ( card `  A
)
-1-1-onto-> A  ->  `' f : A -1-1-onto-> ( card `  A
) )
11 cardon 7757 . . . . . . . 8  |-  ( card `  A )  e.  On
1211onordi 4619 . . . . . . 7  |-  Ord  ( card `  A )
13 ordwe 4528 . . . . . . 7  |-  ( Ord  ( card `  A
)  ->  _E  We  ( card `  A )
)
1412, 13ax-mp 8 . . . . . 6  |-  _E  We  ( card `  A )
15 eqid 2380 . . . . . . 7  |-  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  =  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }
1615f1owe 6005 . . . . . 6  |-  ( `' f : A -1-1-onto-> ( card `  A )  ->  (  _E  We  ( card `  A
)  ->  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  We  A ) )
1710, 14, 16ee10 1382 . . . . 5  |-  ( f : ( card `  A
)
-1-1-onto-> A  ->  { <. z ,  w >.  |  ( `' f `  z
)  _E  ( `' f `  w ) }  We  A )
18 weinxp 4878 . . . . 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 4504 . . . . 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 2973 . . . 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 1643 . 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 1547    e. wcel 1717   _Vcvv 2892    i^i cin 3255   class class class wbr 4146   {copab 4199    _E cep 4426    We wwe 4474   Ord word 4514    X. cxp 4809   `'ccnv 4810   dom cdm 4811   -1-1-onto->wf1o 5386   ` cfv 5387    ~~ cen 7035   cardccrd 7748
This theorem is referenced by:  ween  7842  ac5num  7843  dfac8  7941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-en 7039  df-card 7752
  Copyright terms: Public domain W3C validator