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Theorem ordelon 4891
Description: An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)
Assertion
Ref Expression
ordelon  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  e.  On )

Proof of Theorem ordelon
StepHypRef Expression
1 ordelord 4889 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
2 elong 4875 . . 3  |-  ( B  e.  A  ->  ( B  e.  On  <->  Ord  B ) )
32adantl 464 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  ( B  e.  On  <->  Ord  B ) )
41, 3mpbird 232 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  e.  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1823   Ord word 4866   Oncon0 4867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871
This theorem is referenced by:  onelon  4892  ordunidif  4915  ordpwsuc  6623  ordsucun  6633  ordunel  6635  ordunisuc2  6652  oesuclem  7167  odi  7220  oelim2  7236  oeoalem  7237  oeoelem  7239  limenpsi  7685  ordtypelem9  7943  oismo  7957  cantnflt  8082  cantnfp1lem3  8090  cantnflem1b  8096  cantnflem1  8099  cantnfltOLD  8112  cantnfp1lem3OLD  8116  cantnflem1bOLD  8119  cantnflem1OLD  8122  rankr1bg  8212  rankr1clem  8229  rankr1c  8230  rankonidlem  8237  infxpenlem  8382  coflim  8632  fin23lem26  8696  fpwwe2lem8  9004  nofulllem5  29706  onsuct0  30134
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