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Theorem limelon 4396
Description: A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)
Assertion
Ref Expression
limelon  |-  ( ( A  e.  B  /\  Lim  A )  ->  A  e.  On )

Proof of Theorem limelon
StepHypRef Expression
1 limord 4392 . . 3  |-  ( Lim 
A  ->  Ord  A )
2 elong 4370 . . 3  |-  ( A  e.  B  ->  ( A  e.  On  <->  Ord  A ) )
31, 2syl5ibr 156 . 2  |-  ( A  e.  B  ->  ( Lim  A  ->  A  e.  On ) )
43imp 124 1  |-  ( ( A  e.  B  /\  Lim  A )  ->  A  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2148   Ord word 4359   Oncon0 4360   Lim wlim 4361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-in 3135  df-ss 3142  df-uni 3808  df-tr 4099  df-iord 4363  df-on 4365  df-ilim 4366
This theorem is referenced by: (None)
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