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Theorem onin 4834
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 4813 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 eloni 4813 . . 3  |-  ( B  e.  On  ->  Ord  B )
3 ordin 4833 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
41, 2, 3syl2an 477 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  Ord  ( A  i^i  B ) )
5 simpl 457 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  e.  On )
6 inex1g 4519 . . 3  |-  ( A  e.  On  ->  ( A  i^i  B )  e. 
_V )
7 elong 4811 . . 3  |-  ( ( A  i^i  B )  e.  _V  ->  (
( A  i^i  B
)  e.  On  <->  Ord  ( A  i^i  B ) ) )
85, 6, 73syl 20 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  i^i  B )  e.  On  <->  Ord  ( A  i^i  B ) ) )
94, 8mpbird 232 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  i^i  B
)  e.  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1757   _Vcvv 3054    i^i cin 3411   Ord word 4802   Oncon0 4803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ral 2797  df-rex 2798  df-v 3056  df-in 3419  df-ss 3426  df-uni 4176  df-tr 4470  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807
This theorem is referenced by:  tfrlem5  6925  noreson  27921  ontopbas  28394
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