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Theorem ordeleqon 4580
Description: A way to express the ordinal property of a class in terms of the class of ordinal numbers. Corollary 7.14 of [TakeutiZaring] p. 38 and its converse. (Contributed by NM, 1-Jun-2003.)
Assertion
Ref Expression
ordeleqon  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )

Proof of Theorem ordeleqon
StepHypRef Expression
1 onprc 4576 . . . 4  |-  -.  On  e.  _V
2 elex 2796 . . . 4  |-  ( On  e.  A  ->  On  e.  _V )
31, 2mto 167 . . 3  |-  -.  On  e.  A
4 ordon 4574 . . . . . 6  |-  Ord  On
5 ordtri3or 4424 . . . . . 6  |-  ( ( Ord  A  /\  Ord  On )  ->  ( A  e.  On  \/  A  =  On  \/  On  e.  A ) )
64, 5mpan2 652 . . . . 5  |-  ( Ord 
A  ->  ( A  e.  On  \/  A  =  On  \/  On  e.  A ) )
7 df-3or 935 . . . . 5  |-  ( ( A  e.  On  \/  A  =  On  \/  On  e.  A )  <->  ( ( A  e.  On  \/  A  =  On )  \/  On  e.  A ) )
86, 7sylib 188 . . . 4  |-  ( Ord 
A  ->  ( ( A  e.  On  \/  A  =  On )  \/  On  e.  A ) )
98ord 366 . . 3  |-  ( Ord 
A  ->  ( -.  ( A  e.  On  \/  A  =  On )  ->  On  e.  A
) )
103, 9mt3i 118 . 2  |-  ( Ord 
A  ->  ( A  e.  On  \/  A  =  On ) )
11 eloni 4402 . . 3  |-  ( A  e.  On  ->  Ord  A )
12 ordeq 4399 . . . 4  |-  ( A  =  On  ->  ( Ord  A  <->  Ord  On ) )
134, 12mpbiri 224 . . 3  |-  ( A  =  On  ->  Ord  A )
1411, 13jaoi 368 . 2  |-  ( ( A  e.  On  \/  A  =  On )  ->  Ord  A )
1510, 14impbii 180 1  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    \/ w3o 933    = wceq 1623    e. wcel 1684   _Vcvv 2788   Ord word 4391   Oncon0 4392
This theorem is referenced by:  ordsson  4581  ssonprc  4583  ordunisuc  4623  orduninsuc  4634  limomss  4661  omon  4667  limom  4671  tfrlem14  6407  tfr2b  6412  unialeph  7728  ordtoplem  24874  ordcmp  24886
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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