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Theorem onin 4439
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 4418 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 eloni 4418 . . 3  |-  ( B  e.  On  ->  Ord  B )
3 ordin 4438 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
41, 2, 3syl2an 463 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  Ord  ( A  i^i  B ) )
5 simpl 443 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  e.  On )
6 inex1g 4173 . . 3  |-  ( A  e.  On  ->  ( A  i^i  B )  e. 
_V )
7 elong 4416 . . 3  |-  ( ( A  i^i  B )  e.  _V  ->  (
( A  i^i  B
)  e.  On  <->  Ord  ( A  i^i  B ) ) )
85, 6, 73syl 18 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  i^i  B )  e.  On  <->  Ord  ( A  i^i  B ) ) )
94, 8mpbird 223 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  i^i  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696   _Vcvv 2801    i^i cin 3164   Ord word 4407   Oncon0 4408
This theorem is referenced by:  tfrlem5  6412  noreson  24385  ontopbas  24939
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-uni 3844  df-tr 4130  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
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