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Theorem onin 4278
Description: The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)
Assertion
Ref Expression
onin  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  i^i  B
)  e.  On )

Proof of Theorem onin
StepHypRef Expression
1 eloni 4267 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 eloni 4267 . . 3  |-  ( B  e.  On  ->  Ord  B )
3 ordin 4277 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
41, 2, 3syl2an 287 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  Ord  ( A  i^i  B ) )
5 simpl 108 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  e.  On )
6 inex1g 4034 . . 3  |-  ( A  e.  On  ->  ( A  i^i  B )  e. 
_V )
7 elong 4265 . . 3  |-  ( ( A  i^i  B )  e.  _V  ->  (
( A  i^i  B
)  e.  On  <->  Ord  ( A  i^i  B ) ) )
85, 6, 73syl 17 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  i^i  B )  e.  On  <->  Ord  ( A  i^i  B ) ) )
94, 8mpbird 166 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  i^i  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1465   _Vcvv 2660    i^i cin 3040   Ord word 4254   Oncon0 4255
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-in 3047  df-ss 3054  df-uni 3707  df-tr 3997  df-iord 4258  df-on 4260
This theorem is referenced by:  tfrlem5  6179
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