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Theorem onin 4572
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 4551 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 eloni 4551 . . 3  |-  ( B  e.  On  ->  Ord  B )
3 ordin 4571 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
41, 2, 3syl2an 464 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  Ord  ( A  i^i  B ) )
5 simpl 444 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  e.  On )
6 inex1g 4306 . . 3  |-  ( A  e.  On  ->  ( A  i^i  B )  e. 
_V )
7 elong 4549 . . 3  |-  ( ( A  i^i  B )  e.  _V  ->  (
( A  i^i  B
)  e.  On  <->  Ord  ( A  i^i  B ) ) )
85, 6, 73syl 19 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  i^i  B )  e.  On  <->  Ord  ( A  i^i  B ) ) )
94, 8mpbird 224 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  i^i  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1721   _Vcvv 2916    i^i cin 3279   Ord word 4540   Oncon0 4541
This theorem is referenced by:  tfrlem5  6600  noreson  25528  ontopbas  26082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-v 2918  df-in 3287  df-ss 3294  df-uni 3976  df-tr 4263  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545
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