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Theorem ordelon 4539
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 4537 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
2 elong 4523 . . 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 1717   Ord word 4514   Oncon0 4515
This theorem is referenced by:  onelon  4540  ordunidif  4563  ordpwsuc  4728  ordsucun  4738  ordunel  4740  ordunisuc2  4757  oesuclem  6698  odi  6751  oelim2  6767  oeoalem  6768  oeoelem  6770  limenpsi  7211  ordtypelem9  7421  oismo  7435  cantnflt  7553  cantnfp1lem3  7562  cantnflem1b  7568  cantnflem1  7571  rankr1bg  7655  rankr1clem  7672  rankr1c  7673  rankonidlem  7680  infxpenlem  7821  coflim  8067  fin23lem26  8131  fpwwe2lem8  8438  nofulllem5  25377  onsuct0  25898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-tr 4237  df-eprel 4428  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519
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