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Theorem orduniorsuc 6643
Description: An ordinal class is either its union or the successor of its union. If we adopt the view that zero is a limit ordinal, this means every ordinal class is either a limit or a successor. (Contributed by NM, 13-Sep-2003.)
Assertion
Ref Expression
orduniorsuc  |-  ( Ord 
A  ->  ( A  =  U. A  \/  A  =  suc  U. A ) )

Proof of Theorem orduniorsuc
StepHypRef Expression
1 orduniss 4972 . . . . . 6  |-  ( Ord 
A  ->  U. A  C_  A )
2 orduni 6607 . . . . . . . 8  |-  ( Ord 
A  ->  Ord  U. A
)
3 ordelssne 4905 . . . . . . . 8  |-  ( ( Ord  U. A  /\  Ord  A )  ->  ( U. A  e.  A  <->  ( U. A  C_  A  /\  U. A  =/=  A
) ) )
42, 3mpancom 669 . . . . . . 7  |-  ( Ord 
A  ->  ( U. A  e.  A  <->  ( U. A  C_  A  /\  U. A  =/=  A ) ) )
54biimprd 223 . . . . . 6  |-  ( Ord 
A  ->  ( ( U. A  C_  A  /\  U. A  =/=  A )  ->  U. A  e.  A
) )
61, 5mpand 675 . . . . 5  |-  ( Ord 
A  ->  ( U. A  =/=  A  ->  U. A  e.  A ) )
7 ordsucss 6631 . . . . 5  |-  ( Ord 
A  ->  ( U. A  e.  A  ->  suc  U. A  C_  A ) )
86, 7syld 44 . . . 4  |-  ( Ord 
A  ->  ( U. A  =/=  A  ->  suc  U. A  C_  A )
)
9 ordsucuni 6642 . . . 4  |-  ( Ord 
A  ->  A  C_  suc  U. A )
108, 9jctild 543 . . 3  |-  ( Ord 
A  ->  ( U. A  =/=  A  ->  ( A  C_  suc  U. A  /\  suc  U. A  C_  A ) ) )
11 df-ne 2664 . . . 4  |-  ( A  =/=  U. A  <->  -.  A  =  U. A )
12 necom 2736 . . . 4  |-  ( A  =/=  U. A  <->  U. A  =/= 
A )
1311, 12bitr3i 251 . . 3  |-  ( -.  A  =  U. A  <->  U. A  =/=  A )
14 eqss 3519 . . 3  |-  ( A  =  suc  U. A  <->  ( A  C_  suc  U. A  /\  suc  U. A  C_  A ) )
1510, 13, 143imtr4g 270 . 2  |-  ( Ord 
A  ->  ( -.  A  =  U. A  ->  A  =  suc  U. A
) )
1615orrd 378 1  |-  ( Ord 
A  ->  ( A  =  U. A  \/  A  =  suc  U. A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    C_ wss 3476   U.cuni 4245   Ord word 4877   suc csuc 4880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884
This theorem is referenced by:  onuniorsuci  6652  oeeulem  7247  cantnfp1lem2  8094  cantnflem1  8104  cantnfp1lem2OLD  8120  cantnflem1OLD  8127  cnfcom2lem  8141  cnfcom2lemOLD  8149  dfac12lem1  8519  dfac12lem2  8520  ttukeylem3  8887  ttukeylem5  8889  ttukeylem6  8890  ordtoplem  29477  ordcmp  29489  aomclem5  30608
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