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Theorem onuniorsuci 4786
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 4653 . 2  |-  Ord  A
3 orduniorsuc 4777 . 2  |-  ( Ord 
A  ->  ( A  =  U. A  \/  A  =  suc  U. A ) )
42, 3ax-mp 8 1  |-  ( A  =  U. A  \/  A  =  suc  U. A
)
Colors of variables: wff set class
Syntax hints:    \/ wo 358    = wceq 1649    e. wcel 1721   U.cuni 3983   Ord word 4548   Oncon0 4549   suc csuc 4551
This theorem is referenced by:  onuninsuci  4787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-tr 4271  df-eprel 4462  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-suc 4555
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