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

Theorem nodmon 30365
Description: The domain of a surreal is an ordinal. (Contributed by Scott Fenton, 16-Jun-2011.)
Assertion
Ref Expression
nodmon  |-  ( A  e.  No  ->  dom  A  e.  On )

Proof of Theorem nodmon
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elno 30361 . 2  |-  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } )
2 fdm 5741 . . . . 5  |-  ( A : x --> { 1o ,  2o }  ->  dom  A  =  x )
32eleq1d 2489 . . . 4  |-  ( A : x --> { 1o ,  2o }  ->  ( dom  A  e.  On  <->  x  e.  On ) )
43biimprcd 228 . . 3  |-  ( x  e.  On  ->  ( A : x --> { 1o ,  2o }  ->  dom  A  e.  On ) )
54rexlimiv 2909 . 2  |-  ( E. x  e.  On  A : x --> { 1o ,  2o }  ->  dom  A  e.  On )
61, 5sylbi 198 1  |-  ( A  e.  No  ->  dom  A  e.  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1867   E.wrex 2774   {cpr 3995   dom cdm 4845   Oncon0 5433   -->wf 5588   1oc1o 7174   2oc2o 7175   Nocsur 30355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-no 30358
This theorem is referenced by:  nodmord  30368  elno2  30369  bdayfo  30390  nodenselem5  30400
  Copyright terms: Public domain W3C validator