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Theorem ordelon 4343
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 4341 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  Ord  B )
2 elong 4333 . . 3  |-  ( B  e.  A  ->  ( B  e.  On  <->  Ord  B ) )
32adantl 275 . 2  |-  ( ( Ord  A  /\  B  e.  A )  ->  ( B  e.  On  <->  Ord  B ) )
41, 3mpbird 166 1  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 2128   Ord word 4322   Oncon0 4323
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-in 3108  df-ss 3115  df-uni 3773  df-tr 4063  df-iord 4326  df-on 4328
This theorem is referenced by:  onelon  4344  ordsson  4451  ordpwsucss  4526  tfr1onlemsucfn  6287  tfr1onlemsucaccv  6288  tfr1onlembfn  6291  tfr1onlemubacc  6293  tfr1onlemaccex  6295  tfrcllemsucfn  6300  tfrcllemsucaccv  6301  tfrcllembfn  6304  tfrcllemubacc  6306  tfrcllemaccex  6308  tfrcl  6311
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