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Theorem ordelon 4597
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 4595 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
2 elong 4581 . . 3  |-  ( B  e.  A  ->  ( B  e.  On  <->  Ord  B ) )
32adantl 453 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  ( B  e.  On  <->  Ord  B ) )
41, 3mpbird 224 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725   Ord word 4572   Oncon0 4573
This theorem is referenced by:  onelon  4598  ordunidif  4621  ordpwsuc  4787  ordsucun  4797  ordunel  4799  ordunisuc2  4816  oesuclem  6761  odi  6814  oelim2  6830  oeoalem  6831  oeoelem  6833  limenpsi  7274  ordtypelem9  7487  oismo  7501  cantnflt  7619  cantnfp1lem3  7628  cantnflem1b  7634  cantnflem1  7637  rankr1bg  7721  rankr1clem  7738  rankr1c  7739  rankonidlem  7746  infxpenlem  7887  coflim  8133  fin23lem26  8197  fpwwe2lem8  8504  nofulllem5  25653  onsuct0  26183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577
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