MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  orduniorsuc Unicode version

Theorem orduniorsuc 4621
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 4487 . . . . . 6  |-  ( Ord 
A  ->  U. A  C_  A )
2 orduni 4585 . . . . . . . 8  |-  ( Ord 
A  ->  Ord  U. A
)
3 ordelssne 4419 . . . . . . . 8  |-  ( ( Ord  U. A  /\  Ord  A )  ->  ( U. A  e.  A  <->  ( U. A  C_  A  /\  U. A  =/=  A
) ) )
42, 3mpancom 650 . . . . . . 7  |-  ( Ord 
A  ->  ( U. A  e.  A  <->  ( U. A  C_  A  /\  U. A  =/=  A ) ) )
54biimprd 214 . . . . . 6  |-  ( Ord 
A  ->  ( ( U. A  C_  A  /\  U. A  =/=  A )  ->  U. A  e.  A
) )
61, 5mpand 656 . . . . 5  |-  ( Ord 
A  ->  ( U. A  =/=  A  ->  U. A  e.  A ) )
7 ordsucss 4609 . . . . 5  |-  ( Ord 
A  ->  ( U. A  e.  A  ->  suc  U. A  C_  A ) )
86, 7syld 40 . . . 4  |-  ( Ord 
A  ->  ( U. A  =/=  A  ->  suc  U. A  C_  A )
)
9 ordsucuni 4620 . . . 4  |-  ( Ord 
A  ->  A  C_  suc  U. A )
108, 9jctild 527 . . 3  |-  ( Ord 
A  ->  ( U. A  =/=  A  ->  ( A  C_  suc  U. A  /\  suc  U. A  C_  A ) ) )
11 df-ne 2448 . . . 4  |-  ( A  =/=  U. A  <->  -.  A  =  U. A )
12 necom 2527 . . . 4  |-  ( A  =/=  U. A  <->  U. A  =/= 
A )
1311, 12bitr3i 242 . . 3  |-  ( -.  A  =  U. A  <->  U. A  =/=  A )
14 eqss 3194 . . 3  |-  ( A  =  suc  U. A  <->  ( A  C_  suc  U. A  /\  suc  U. A  C_  A ) )
1510, 13, 143imtr4g 261 . 2  |-  ( Ord 
A  ->  ( -.  A  =  U. A  ->  A  =  suc  U. A
) )
1615orrd 367 1  |-  ( Ord 
A  ->  ( A  =  U. A  \/  A  =  suc  U. A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   U.cuni 3827   Ord word 4391   suc csuc 4394
This theorem is referenced by:  onuniorsuci  4630  oeeulem  6599  cantnfp1lem2  7381  cantnflem1  7391  cnfcom2lem  7404  dfac12lem1  7769  dfac12lem2  7770  ttukeylem3  8138  ttukeylem5  8140  ttukeylem6  8141  ordtoplem  24285  ordcmp  24297  aomclem5  26567
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398
  Copyright terms: Public domain W3C validator