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Theorem axbday 23683
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

Proof of Theorem axbday
StepHypRef Expression
1 dmexg 4913 . . . 4  |-  ( x  e.  No  ->  dom  x  e.  _V )
21rgen 2581 . . 3  |-  A. x  e.  No  dom  x  e. 
_V
3 df-bday 23654 . . . 4  |-  bday  =  ( x  e.  No  |->  dom  x )
43mptfng 5293 . . 3  |-  ( A. x  e.  No  dom  x  e.  _V  <->  bday  Fn  No )
52, 4mpbi 201 . 2  |-  bday  Fn  No
63rnmpt 4899 . . 3  |-  ran  bday  =  { y  |  E. x  e.  No  y  =  dom  x }
7 noxp1o 23674 . . . . . 6  |-  ( y  e.  On  ->  (
y  X.  { 1o } )  e.  No )
8 1on 6440 . . . . . . . . . 10  |-  1o  e.  On
98elexi 2766 . . . . . . . . 9  |-  1o  e.  _V
109snnz 3704 . . . . . . . 8  |-  { 1o }  =/=  (/)
11 dmxp 4871 . . . . . . . 8  |-  ( { 1o }  =/=  (/)  ->  dom  (  y  X.  { 1o } )  =  y )
1210, 11ax-mp 10 . . . . . . 7  |-  dom  ( 
y  X.  { 1o } )  =  y
1312eqcomi 2260 . . . . . 6  |-  y  =  dom  (  y  X. 
{ 1o } )
14 dmeq 4853 . . . . . . . 8  |-  ( x  =  ( y  X. 
{ 1o } )  ->  dom  x  =  dom  (  y  X.  { 1o } ) )
1514eqeq2d 2267 . . . . . . 7  |-  ( x  =  ( y  X. 
{ 1o } )  ->  ( y  =  dom  x  <->  y  =  dom  (  y  X.  { 1o } ) ) )
1615rcla4ev 2852 . . . . . 6  |-  ( ( ( y  X.  { 1o } )  e.  No  /\  y  =  dom  ( 
y  X.  { 1o } ) )  ->  E. x  e.  No  y  =  dom  x )
177, 13, 16sylancl 646 . . . . 5  |-  ( y  e.  On  ->  E. x  e.  No  y  =  dom  x )
18 nodmon 23659 . . . . . . 7  |-  ( x  e.  No  ->  dom  x  e.  On )
19 eleq1a 2325 . . . . . . 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 2634 . . . . 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 2367 . . 3  |-  On  =  { y  |  E. x  e.  No  y  =  dom  x }
246, 23eqtr4i 2279 . 2  |-  ran  bday  =  On
25 df-fo 4673 . 2  |-  ( bday
: No -onto-> On  <->  ( bday  Fn  No  /\  ran  bday  =  On ) )
265, 24, 25mpbir2an 891 1  |-  bday : No -onto-> On
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621   {cab 2242    =/= wne 2419   A.wral 2516   E.wrex 2517   _Vcvv 2757   (/)c0 3416   {csn 3600   Oncon0 4350    X. cxp 4645   dom cdm 4647   ran crn 4648    Fn wfn 4654   -onto->wfo 4657   1oc1o 6426   Nocsur 23649   bdaycbday 23651
This theorem is referenced by:  bdayfun  23684  bdayrn  23685  bdaydm  23686  noprc  23689
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-suc 4356  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-1o 6433  df-no 23652  df-bday 23654
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