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Theorem onuniorsuci 4848
Description: An ordinal number is either its own union (if zero or a limit ordinal) or the successor of its union. (Contributed by NM, 13-Jun-1994.)
Hypothesis
Ref Expression
onssi.1  |-  A  e.  On
Assertion
Ref Expression
onuniorsuci  |-  ( A  =  U. A  \/  A  =  suc  U. A
)

Proof of Theorem onuniorsuci
StepHypRef Expression
1 onssi.1 . . 3  |-  A  e.  On
21onordi 4715 . 2  |-  Ord  A
3 orduniorsuc 4839 . 2  |-  ( Ord 
A  ->  ( A  =  U. A  \/  A  =  suc  U. A ) )
42, 3ax-mp 5 1  |-  ( A  =  U. A  \/  A  =  suc  U. A
)
Colors of variables: wff set class
Syntax hints:    \/ wo 359    = wceq 1653    e. wcel 1727   U.cuni 4039   Ord word 4609   Oncon0 4610   suc csuc 4612
This theorem is referenced by:  onuninsuci  4849
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 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432  ax-un 4730
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-tr 4328  df-eprel 4523  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-suc 4616
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