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Theorem onin 4222
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 4211 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 eloni 4211 . . 3  |-  ( B  e.  On  ->  Ord  B )
3 ordin 4221 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
41, 2, 3syl2an 284 . 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 3981 . . 3  |-  ( A  e.  On  ->  ( A  i^i  B )  e. 
_V )
7 elong 4209 . . 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 1439   _Vcvv 2620    i^i cin 2999   Ord word 4198   Oncon0 4199
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-in 3006  df-ss 3013  df-uni 3660  df-tr 3943  df-iord 4202  df-on 4204
This theorem is referenced by:  tfrlem5  6093
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