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Theorem ordzsl 4828
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 4826 . . . . . 6  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
21biimprd 216 . . . . 5  |-  ( Ord 
A  ->  ( -.  E. x  e.  On  A  =  suc  x  ->  A  =  U. A ) )
3 unizlim 4701 . . . . 5  |-  ( Ord 
A  ->  ( A  =  U. A  <->  ( A  =  (/)  \/  Lim  A
) ) )
42, 3sylibd 207 . . . 4  |-  ( Ord 
A  ->  ( -.  E. x  e.  On  A  =  suc  x  ->  ( A  =  (/)  \/  Lim  A ) ) )
54orrd 369 . . 3  |-  ( Ord 
A  ->  ( E. x  e.  On  A  =  suc  x  \/  ( A  =  (/)  \/  Lim  A ) ) )
6 3orass 940 . . . 4  |-  ( ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A )  <->  ( A  =  (/)  \/  ( E. x  e.  On  A  =  suc  x  \/  Lim  A ) ) )
7 or12 511 . . . 4  |-  ( ( A  =  (/)  \/  ( E. x  e.  On  A  =  suc  x  \/ 
Lim  A ) )  <-> 
( E. x  e.  On  A  =  suc  x  \/  ( A  =  (/)  \/  Lim  A
) ) )
86, 7bitri 242 . . 3  |-  ( ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A )  <->  ( E. x  e.  On  A  =  suc  x  \/  ( A  =  (/)  \/  Lim  A
) ) )
95, 8sylibr 205 . 2  |-  ( Ord 
A  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A ) )
10 ord0 4636 . . . 4  |-  Ord  (/)
11 ordeq 4591 . . . 4  |-  ( A  =  (/)  ->  ( Ord 
A  <->  Ord  (/) ) )
1210, 11mpbiri 226 . . 3  |-  ( A  =  (/)  ->  Ord  A
)
13 suceloni 4796 . . . . . 6  |-  ( x  e.  On  ->  suc  x  e.  On )
14 eleq1 2498 . . . . . 6  |-  ( A  =  suc  x  -> 
( A  e.  On  <->  suc  x  e.  On ) )
1513, 14syl5ibr 214 . . . . 5  |-  ( A  =  suc  x  -> 
( x  e.  On  ->  A  e.  On ) )
16 eloni 4594 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
1715, 16syl6com 34 . . . 4  |-  ( x  e.  On  ->  ( A  =  suc  x  ->  Ord  A ) )
1817rexlimiv 2826 . . 3  |-  ( E. x  e.  On  A  =  suc  x  ->  Ord  A )
19 limord 4643 . . 3  |-  ( Lim 
A  ->  Ord  A )
2012, 18, 193jaoi 1248 . 2  |-  ( ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A )  ->  Ord  A )
219, 20impbii 182 1  |-  ( Ord 
A  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    \/ wo 359    \/ w3o 936    = wceq 1653    e. wcel 1726   E.wrex 2708   (/)c0 3630   U.cuni 4017   Ord word 4583   Oncon0 4584   Lim wlim 4585   suc csuc 4586
This theorem is referenced by:  onzsl  4829  tfrlem16  6657  omeulem1  6828  oaabs2  6891  rankxplim3  7810  rankxpsuc  7811  cardlim  7864  cardaleph  7975  cflim2  8148  dfrdg2  25428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-tr 4306  df-eprel 4497  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590
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