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Theorem axbday 23496
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 4846 . . . 4  |-  ( x  e.  No  ->  dom  x  e.  _V )
21rgen 2570 . . 3  |-  A. x  e.  No  dom  x  e. 
_V
3 df-bday 23467 . . . 4  |-  bday  =  ( x  e.  No  |->  dom  x )
43mptfng 5226 . . 3  |-  ( A. x  e.  No  dom  x  e.  _V  <->  bday  Fn  No )
52, 4mpbi 201 . 2  |-  bday  Fn  No
63rnmpt 4832 . . 3  |-  ran  bday  =  { y  |  E. x  e.  No  y  =  dom  x }
7 noxp1o 23487 . . . . . 6  |-  ( y  e.  On  ->  (
y  X.  { 1o } )  e.  No )
8 1on 6372 . . . . . . . . . 10  |-  1o  e.  On
98elexi 2736 . . . . . . . . 9  |-  1o  e.  _V
109snnz 3648 . . . . . . . 8  |-  { 1o }  =/=  (/)
11 dmxp 4804 . . . . . . . 8  |-  ( { 1o }  =/=  (/)  ->  dom  (  y  X.  { 1o } )  =  y )
1210, 11ax-mp 10 . . . . . . 7  |-  dom  ( 
y  X.  { 1o } )  =  y
1312eqcomi 2257 . . . . . 6  |-  y  =  dom  (  y  X. 
{ 1o } )
14 dmeq 4786 . . . . . . . 8  |-  ( x  =  ( y  X. 
{ 1o } )  ->  dom  x  =  dom  (  y  X.  { 1o } ) )
1514eqeq2d 2264 . . . . . . 7  |-  ( x  =  ( y  X. 
{ 1o } )  ->  ( y  =  dom  x  <->  y  =  dom  (  y  X.  { 1o } ) ) )
1615rcla4ev 2821 . . . . . 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 23472 . . . . . . 7  |-  ( x  e.  No  ->  dom  x  e.  On )
19 eleq1a 2322 . . . . . . 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 2623 . . . . 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 2360 . . 3  |-  On  =  { y  |  E. x  e.  No  y  =  dom  x }
246, 23eqtr4i 2276 . 2  |-  ran  bday  =  On
25 df-fo 4606 . 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 2239    =/= wne 2412   A.wral 2509   E.wrex 2510   _Vcvv 2727   (/)c0 3362   {csn 3544   Oncon0 4285    X. cxp 4578   dom cdm 4580   ran crn 4581    Fn wfn 4587   -onto->wfo 4590   1oc1o 6358   Nocsur 23462   bdaycbday 23464
This theorem is referenced by:  bdayfun  23497  bdayrn  23498  bdaydm  23499  noprc  23502
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-1o 6365  df-no 23465  df-bday 23467
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