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Theorem axbday 23730
Description: The birthday function maps the surreals onto the ordinals. Alling's axiom (B). (Shortened proof on 2012-Apr-14, SF). (Contributed by Scott Fenton, 11-Jun-2011.)
Assertion
Ref Expression
axbday  |-  bday : No -onto-> On
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem axbday
StepHypRef Expression
1 dmexg 4939 . . . 4  |-  ( x  e.  No  ->  dom  x  e.  _V )
21rgen 2610 . . 3  |-  A. x  e.  No  dom  x  e. 
_V
3 df-bday 23701 . . . 4  |-  bday  =  ( x  e.  No  |->  dom  x )
43mptfng 5335 . . 3  |-  ( A. x  e.  No  dom  x  e.  _V  <->  bday  Fn  No )
52, 4mpbi 201 . 2  |-  bday  Fn  No
63rnmpt 4925 . . 3  |-  ran  bday  =  { y  |  E. x  e.  No  y  =  dom  x }
7 noxp1o 23721 . . . . . 6  |-  ( y  e.  On  ->  (
y  X.  { 1o } )  e.  No )
8 1on 6482 . . . . . . . . . 10  |-  1o  e.  On
98elexi 2799 . . . . . . . . 9  |-  1o  e.  _V
109snnz 3746 . . . . . . . 8  |-  { 1o }  =/=  (/)
11 dmxp 4897 . . . . . . . 8  |-  ( { 1o }  =/=  (/)  ->  dom  (  y  X.  { 1o } )  =  y )
1210, 11ax-mp 10 . . . . . . 7  |-  dom  ( 
y  X.  { 1o } )  =  y
1312eqcomi 2289 . . . . . 6  |-  y  =  dom  (  y  X. 
{ 1o } )
14 dmeq 4879 . . . . . . . 8  |-  ( x  =  ( y  X. 
{ 1o } )  ->  dom  x  =  dom  (  y  X.  { 1o } ) )
1514eqeq2d 2296 . . . . . . 7  |-  ( x  =  ( y  X. 
{ 1o } )  ->  ( y  =  dom  x  <->  y  =  dom  (  y  X.  { 1o } ) ) )
1615rspcev 2886 . . . . . 6  |-  ( ( ( y  X.  { 1o } )  e.  No  /\  y  =  dom  ( 
y  X.  { 1o } ) )  ->  E. x  e.  No  y  =  dom  x )
177, 13, 16sylancl 645 . . . . 5  |-  ( y  e.  On  ->  E. x  e.  No  y  =  dom  x )
18 nodmon 23706 . . . . . . 7  |-  ( x  e.  No  ->  dom  x  e.  On )
19 eleq1a 2354 . . . . . . 7  |-  ( dom  x  e.  On  ->  ( y  =  dom  x  ->  y  e.  On ) )
2018, 19syl 17 . . . . . 6  |-  ( x  e.  No  ->  (
y  =  dom  x  ->  y  e.  On ) )
2120rexlimiv 2663 . . . . 5  |-  ( E. x  e.  No  y  =  dom  x  ->  y  e.  On )
2217, 21impbii 182 . . . 4  |-  ( y  e.  On  <->  E. x  e.  No  y  =  dom  x )
2322abbi2i 2396 . . 3  |-  On  =  { y  |  E. x  e.  No  y  =  dom  x }
246, 23eqtr4i 2308 . 2  |-  ran  bday  =  On
25 df-fo 5228 . 2  |-  ( bday
: No -onto-> On  <->  ( bday  Fn  No  /\  ran  bday  =  On ) )
265, 24, 25mpbir2an 888 1  |-  bday : No -onto-> On
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624    e. wcel 1685   {cab 2271    =/= wne 2448   A.wral 2545   E.wrex 2546   _Vcvv 2790   (/)c0 3457   {csn 3642   Oncon0 4392    X. cxp 4687   dom cdm 4689   ran crn 4690    Fn wfn 5217   -onto->wfo 5220   1oc1o 6468   Nocsur 23696   bdaycbday 23698
This theorem is referenced by:  bdayfun  23731  bdayrn  23732  bdaydm  23733  noprc  23736
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  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-suc 4398  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-1o 6475  df-no 23699  df-bday 23701
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