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Theorem ordeleqon 4552
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 4548 . . . 4  |-  -.  On  e.  _V
2 elex 2771 . . . 4  |-  ( On  e.  A  ->  On  e.  _V )
31, 2mto 169 . . 3  |-  -.  On  e.  A
4 ordon 4546 . . . . . 6  |-  Ord  On
5 ordtri3or 4396 . . . . . 6  |-  ( ( Ord  A  /\  Ord  On )  ->  ( A  e.  On  \/  A  =  On  \/  On  e.  A ) )
64, 5mpan2 655 . . . . 5  |-  ( Ord 
A  ->  ( A  e.  On  \/  A  =  On  \/  On  e.  A ) )
7 df-3or 940 . . . . 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 4374 . . 3  |-  ( A  e.  On  ->  Ord  A )
12 ordeq 4371 . . . 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 938    = wceq 1619    e. wcel 1621   _Vcvv 2763   Ord word 4363   Oncon0 4364
This theorem is referenced by:  ordsson  4553  ssonprc  4555  ordunisuc  4595  orduninsuc  4606  limomss  4633  omon  4639  limom  4643  tfrlem14  6375  tfr2b  6380  unialeph  7696  ordtoplem  24249  ordcmp  24261
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-tr 4088  df-eprel 4277  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368
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