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Theorem ordelon 4432
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 4430 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
2 elong 4416 . . 3  |-  ( B  e.  A  ->  ( B  e.  On  <->  Ord  B ) )
32adantl 452 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  ( B  e.  On  <->  Ord  B ) )
41, 3mpbird 223 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696   Ord word 4407   Oncon0 4408
This theorem is referenced by:  onelon  4433  ordunidif  4456  ordpwsuc  4622  ordsucun  4632  ordunel  4634  ordunisuc2  4651  oesuclem  6540  odi  6593  oelim2  6609  oeoalem  6610  oeoelem  6612  limenpsi  7052  ordtypelem9  7257  oismo  7271  cantnflt  7389  cantnfp1lem3  7398  cantnflem1b  7404  cantnflem1  7407  rankr1bg  7491  rankr1clem  7508  rankr1c  7509  rankonidlem  7516  infxpenlem  7657  coflim  7903  fin23lem26  7967  fpwwe2lem8  8275  nofulllem5  24431  onsuct0  24952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
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