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Theorem nodmon 29263
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 29259 . 2  |-  ( A  e.  No  <->  E. x  e.  On  A : x --> { 1o ,  2o } )
2 fdm 5735 . . . . 5  |-  ( A : x --> { 1o ,  2o }  ->  dom  A  =  x )
32eleq1d 2536 . . . 4  |-  ( A : x --> { 1o ,  2o }  ->  ( dom  A  e.  On  <->  x  e.  On ) )
43biimprcd 225 . . 3  |-  ( x  e.  On  ->  ( A : x --> { 1o ,  2o }  ->  dom  A  e.  On ) )
54rexlimiv 2949 . 2  |-  ( E. x  e.  On  A : x --> { 1o ,  2o }  ->  dom  A  e.  On )
61, 5sylbi 195 1  |-  ( A  e.  No  ->  dom  A  e.  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   E.wrex 2815   {cpr 4029   Oncon0 4878   dom cdm 4999   -->wf 5584   1oc1o 7124   2oc2o 7125   Nocsur 29253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-no 29256
This theorem is referenced by:  nodmord  29266  elno2  29267  bdayfo  29288  nodenselem5  29298
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