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Theorem onuniorsuci 6547
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 4918 . 2  |-  Ord  A
3 orduniorsuc 6538 . 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 setvar class
Syntax hints:    \/ wo 368    = wceq 1370    e. wcel 1758   U.cuni 4186   Ord word 4813   Oncon0 4814   suc csuc 4816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-tr 4481  df-eprel 4727  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-suc 4820
This theorem is referenced by:  onuninsuci  6548
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