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

Theorem onuniorsuci 4733
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 4600 . 2  |-  Ord  A
3 orduniorsuc 4724 . 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 357    = wceq 1647    e. wcel 1715   U.cuni 3929   Ord word 4494   Oncon0 4495   suc csuc 4497
This theorem is referenced by:  onuninsuci  4734
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-tr 4216  df-eprel 4408  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-suc 4501
  Copyright terms: Public domain W3C validator