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Theorem nodmon 24862
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 24858 . 2  |-  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } )
2 fdm 5476 . . . . 5  |-  ( A : x --> { 1o ,  2o }  ->  dom  A  =  x )
32eleq1d 2424 . . . 4  |-  ( A : x --> { 1o ,  2o }  ->  ( dom  A  e.  On  <->  x  e.  On ) )
43biimprcd 216 . . 3  |-  ( x  e.  On  ->  ( A : x --> { 1o ,  2o }  ->  dom  A  e.  On ) )
54rexlimiv 2737 . 2  |-  ( E. x  e.  On  A : x --> { 1o ,  2o }  ->  dom  A  e.  On )
61, 5sylbi 187 1  |-  ( A  e.  No  ->  dom  A  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1710   E.wrex 2620   {cpr 3717   Oncon0 4474   dom cdm 4771   -->wf 5333   1oc1o 6559   2oc2o 6560   Nocsur 24852
This theorem is referenced by:  nodmord  24865  elno2  24866  bdayfo  24887  nodenselem5  24897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-no 24855
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