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Theorem ordeleqon 6511
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 6507 . . . 4  |-  -.  On  e.  _V
2 elex 3087 . . . 4  |-  ( On  e.  A  ->  On  e.  _V )
31, 2mto 176 . . 3  |-  -.  On  e.  A
4 ordon 6505 . . . . . 6  |-  Ord  On
5 ordtri3or 4860 . . . . . 6  |-  ( ( Ord  A  /\  Ord  On )  ->  ( A  e.  On  \/  A  =  On  \/  On  e.  A ) )
64, 5mpan2 671 . . . . 5  |-  ( Ord 
A  ->  ( A  e.  On  \/  A  =  On  \/  On  e.  A ) )
7 df-3or 966 . . . . 5  |-  ( ( A  e.  On  \/  A  =  On  \/  On  e.  A )  <->  ( ( A  e.  On  \/  A  =  On )  \/  On  e.  A ) )
86, 7sylib 196 . . . 4  |-  ( Ord 
A  ->  ( ( A  e.  On  \/  A  =  On )  \/  On  e.  A ) )
98ord 377 . . 3  |-  ( Ord 
A  ->  ( -.  ( A  e.  On  \/  A  =  On )  ->  On  e.  A
) )
103, 9mt3i 126 . 2  |-  ( Ord 
A  ->  ( A  e.  On  \/  A  =  On ) )
11 eloni 4838 . . 3  |-  ( A  e.  On  ->  Ord  A )
12 ordeq 4835 . . . 4  |-  ( A  =  On  ->  ( Ord  A  <->  Ord  On ) )
134, 12mpbiri 233 . . 3  |-  ( A  =  On  ->  Ord  A )
1411, 13jaoi 379 . 2  |-  ( ( A  e.  On  \/  A  =  On )  ->  Ord  A )
1510, 14impbii 188 1  |-  ( Ord 
A  <->  ( A  e.  On  \/  A  =  On ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    \/ w3o 964    = wceq 1370    e. wcel 1758   _Vcvv 3078   Ord word 4827   Oncon0 4828
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-tr 4495  df-eprel 4741  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832
This theorem is referenced by:  ordsson  6512  ssonprc  6514  ordunisuc  6554  orduninsuc  6565  limomss  6592  omon  6598  limom  6602  tfrlem14  6961  tfr2b  6966  unialeph  8383  ordtoplem  28426  ordcmp  28438
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