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Theorem ordeleqon 4581
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 4577 . . . 4  |-  -.  On  e.  _V
2 elex 2799 . . . 4  |-  ( On  e.  A  ->  On  e.  _V )
31, 2mto 169 . . 3  |-  -.  On  e.  A
4 ordon 4575 . . . . . 6  |-  Ord  On
5 ordtri3or 4425 . . . . . 6  |-  ( ( Ord  A  /\  Ord  On )  ->  ( A  e.  On  \/  A  =  On  \/  On  e.  A ) )
64, 5mpan2 654 . . . . 5  |-  ( Ord 
A  ->  ( A  e.  On  \/  A  =  On  \/  On  e.  A ) )
7 df-3or 937 . . . . 5  |-  ( ( A  e.  On  \/  A  =  On  \/  On  e.  A )  <->  ( ( A  e.  On  \/  A  =  On )  \/  On  e.  A ) )
86, 7sylib 190 . . . 4  |-  ( Ord 
A  ->  ( ( A  e.  On  \/  A  =  On )  \/  On  e.  A ) )
98ord 368 . . 3  |-  ( Ord 
A  ->  ( -.  ( A  e.  On  \/  A  =  On )  ->  On  e.  A
) )
103, 9mt3i 120 . 2  |-  ( Ord 
A  ->  ( A  e.  On  \/  A  =  On ) )
11 eloni 4403 . . 3  |-  ( A  e.  On  ->  Ord  A )
12 ordeq 4400 . . . 4  |-  ( A  =  On  ->  ( Ord  A  <->  Ord  On ) )
134, 12mpbiri 226 . . 3  |-  ( A  =  On  ->  Ord  A )
1411, 13jaoi 370 . 2  |-  ( ( A  e.  On  \/  A  =  On )  ->  Ord  A )
1510, 14impbii 182 1  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    \/ wo 359    \/ w3o 935    = wceq 1625    e. wcel 1687   _Vcvv 2791   Ord word 4392   Oncon0 4393
This theorem is referenced by:  ordsson  4582  ssonprc  4584  ordunisuc  4624  orduninsuc  4635  limomss  4662  omon  4668  limom  4672  tfrlem14  6404  tfr2b  6409  unialeph  7725  ordtoplem  24283  ordcmp  24295
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-sep 4144  ax-nul 4152  ax-pr 4215  ax-un 4513
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-ral 2551  df-rex 2552  df-rab 2555  df-v 2793  df-sbc 2995  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-br 4027  df-opab 4081  df-tr 4117  df-eprel 4306  df-po 4315  df-so 4316  df-fr 4353  df-we 4355  df-ord 4396  df-on 4397
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