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Theorem ordzsl 4784
Description: An ordinal is zero, a successor ordinal, or a limit ordinal. (Contributed by NM, 1-Oct-2003.)
Assertion
Ref Expression
ordzsl  |-  ( Ord 
A  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A ) )
Distinct variable group:    x, A

Proof of Theorem ordzsl
StepHypRef Expression
1 orduninsuc 4782 . . . . . 6  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
21biimprd 215 . . . . 5  |-  ( Ord 
A  ->  ( -.  E. x  e.  On  A  =  suc  x  ->  A  =  U. A ) )
3 unizlim 4657 . . . . 5  |-  ( Ord 
A  ->  ( A  =  U. A  <->  ( A  =  (/)  \/  Lim  A
) ) )
42, 3sylibd 206 . . . 4  |-  ( Ord 
A  ->  ( -.  E. x  e.  On  A  =  suc  x  ->  ( A  =  (/)  \/  Lim  A ) ) )
54orrd 368 . . 3  |-  ( Ord 
A  ->  ( E. x  e.  On  A  =  suc  x  \/  ( A  =  (/)  \/  Lim  A ) ) )
6 3orass 939 . . . 4  |-  ( ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A )  <->  ( A  =  (/)  \/  ( E. x  e.  On  A  =  suc  x  \/  Lim  A ) ) )
7 or12 510 . . . 4  |-  ( ( A  =  (/)  \/  ( E. x  e.  On  A  =  suc  x  \/ 
Lim  A ) )  <-> 
( E. x  e.  On  A  =  suc  x  \/  ( A  =  (/)  \/  Lim  A
) ) )
86, 7bitri 241 . . 3  |-  ( ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A )  <->  ( E. x  e.  On  A  =  suc  x  \/  ( A  =  (/)  \/  Lim  A
) ) )
95, 8sylibr 204 . 2  |-  ( Ord 
A  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A ) )
10 ord0 4593 . . . 4  |-  Ord  (/)
11 ordeq 4548 . . . 4  |-  ( A  =  (/)  ->  ( Ord 
A  <->  Ord  (/) ) )
1210, 11mpbiri 225 . . 3  |-  ( A  =  (/)  ->  Ord  A
)
13 suceloni 4752 . . . . . 6  |-  ( x  e.  On  ->  suc  x  e.  On )
14 eleq1 2464 . . . . . 6  |-  ( A  =  suc  x  -> 
( A  e.  On  <->  suc  x  e.  On ) )
1513, 14syl5ibr 213 . . . . 5  |-  ( A  =  suc  x  -> 
( x  e.  On  ->  A  e.  On ) )
16 eloni 4551 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
1715, 16syl6com 33 . . . 4  |-  ( x  e.  On  ->  ( A  =  suc  x  ->  Ord  A ) )
1817rexlimiv 2784 . . 3  |-  ( E. x  e.  On  A  =  suc  x  ->  Ord  A )
19 limord 4600 . . 3  |-  ( Lim 
A  ->  Ord  A )
2012, 18, 193jaoi 1247 . 2  |-  ( ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A )  ->  Ord  A )
219, 20impbii 181 1  |-  ( Ord 
A  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    \/ w3o 935    = wceq 1649    e. wcel 1721   E.wrex 2667   (/)c0 3588   U.cuni 3975   Ord word 4540   Oncon0 4541   Lim wlim 4542   suc csuc 4543
This theorem is referenced by:  onzsl  4785  tfrlem16  6613  omeulem1  6784  oaabs2  6847  rankxplim3  7761  rankxpsuc  7762  cardlim  7815  cardaleph  7926  cflim2  8099  dfrdg2  25366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547
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