Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bdayfo Structured version   Unicode version

Theorem bdayfo 27838
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
bdayfo  |-  bday : No -onto-> On

Proof of Theorem bdayfo
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmexg 6530 . . . 4  |-  ( x  e.  No  ->  dom  x  e.  _V )
21rgen 2802 . . 3  |-  A. x  e.  No  dom  x  e. 
_V
3 df-bday 27808 . . . 4  |-  bday  =  ( x  e.  No  |->  dom  x )
43mptfng 5557 . . 3  |-  ( A. x  e.  No  dom  x  e.  _V  <->  bday  Fn  No )
52, 4mpbi 208 . 2  |-  bday  Fn  No
63rnmpt 5106 . . 3  |-  ran  bday  =  { y  |  E. x  e.  No  y  =  dom  x }
7 noxp1o 27829 . . . . . 6  |-  ( y  e.  On  ->  (
y  X.  { 1o } )  e.  No )
8 1on 6948 . . . . . . . . . 10  |-  1o  e.  On
98elexi 3003 . . . . . . . . 9  |-  1o  e.  _V
109snnz 4014 . . . . . . . 8  |-  { 1o }  =/=  (/)
11 dmxp 5079 . . . . . . . 8  |-  ( { 1o }  =/=  (/)  ->  dom  ( y  X.  { 1o } )  =  y )
1210, 11ax-mp 5 . . . . . . 7  |-  dom  (
y  X.  { 1o } )  =  y
1312eqcomi 2447 . . . . . 6  |-  y  =  dom  ( y  X. 
{ 1o } )
14 dmeq 5061 . . . . . . . 8  |-  ( x  =  ( y  X. 
{ 1o } )  ->  dom  x  =  dom  ( y  X.  { 1o } ) )
1514eqeq2d 2454 . . . . . . 7  |-  ( x  =  ( y  X. 
{ 1o } )  ->  ( y  =  dom  x  <->  y  =  dom  ( y  X.  { 1o } ) ) )
1615rspcev 3094 . . . . . 6  |-  ( ( ( y  X.  { 1o } )  e.  No  /\  y  =  dom  (
y  X.  { 1o } ) )  ->  E. x  e.  No  y  =  dom  x )
177, 13, 16sylancl 662 . . . . 5  |-  ( y  e.  On  ->  E. x  e.  No  y  =  dom  x )
18 nodmon 27813 . . . . . . 7  |-  ( x  e.  No  ->  dom  x  e.  On )
19 eleq1a 2512 . . . . . . 7  |-  ( dom  x  e.  On  ->  ( y  =  dom  x  ->  y  e.  On ) )
2018, 19syl 16 . . . . . 6  |-  ( x  e.  No  ->  (
y  =  dom  x  ->  y  e.  On ) )
2120rexlimiv 2856 . . . . 5  |-  ( E. x  e.  No  y  =  dom  x  ->  y  e.  On )
2217, 21impbii 188 . . . 4  |-  ( y  e.  On  <->  E. x  e.  No  y  =  dom  x )
2322abbi2i 2560 . . 3  |-  On  =  { y  |  E. x  e.  No  y  =  dom  x }
246, 23eqtr4i 2466 . 2  |-  ran  bday  =  On
25 df-fo 5445 . 2  |-  ( bday
: No -onto-> On  <->  ( bday  Fn  No  /\  ran  bday  =  On ) )
265, 24, 25mpbir2an 911 1  |-  bday : No -onto-> On
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   {cab 2429    =/= wne 2620   A.wral 2736   E.wrex 2737   _Vcvv 2993   (/)c0 3658   {csn 3898   Oncon0 4740    X. cxp 4859   dom cdm 4861   ran crn 4862    Fn wfn 5434   -onto->wfo 5437   1oc1o 6934   Nocsur 27803   bdaycbday 27805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-1o 6941  df-no 27806  df-bday 27808
This theorem is referenced by:  bdayfun  27839  bdayrn  27840  bdaydm  27841  noprc  27844
  Copyright terms: Public domain W3C validator