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Theorem onelon 4276
Description: An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.)
Assertion
Ref Expression
onelon  |-  ( ( A  e.  On  /\  B  e.  A )  ->  B  e.  On )

Proof of Theorem onelon
StepHypRef Expression
1 eloni 4267 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ordelon 4275 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  e.  On )
31, 2sylan 281 1  |-  ( ( A  e.  On  /\  B  e.  A )  ->  B  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1465   Ord word 4254   Oncon0 4255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-in 3047  df-ss 3054  df-uni 3707  df-tr 3997  df-iord 4258  df-on 4260
This theorem is referenced by:  oneli  4320  ssorduni  4373  unon  4397  tfrlemibacc  6191  tfrlemibxssdm  6192  tfrlemibfn  6193  tfrexlem  6199  tfr1onlemsucaccv  6206  tfrcllemsucaccv  6219  sucinc2  6310  oav2  6327  omv2  6329
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