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Theorem ordeleqon 4617
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 4613 . . . 4  |-  -.  On  e.  _V
2 elex 2830 . . . 4  |-  ( On  e.  A  ->  On  e.  _V )
31, 2mto 167 . . 3  |-  -.  On  e.  A
4 ordon 4611 . . . . . 6  |-  Ord  On
5 ordtri3or 4461 . . . . . 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 4439 . . 3  |-  ( A  e.  On  ->  Ord  A )
12 ordeq 4436 . . . 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 1633    e. wcel 1701   _Vcvv 2822   Ord word 4428   Oncon0 4429
This theorem is referenced by:  ordsson  4618  ssonprc  4620  ordunisuc  4660  orduninsuc  4671  limomss  4698  omon  4704  limom  4708  tfrlem14  6449  tfr2b  6454  unialeph  7773  ordtoplem  25260  ordcmp  25272
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-tr 4151  df-eprel 4342  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433
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