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

Theorem onuniorsuci 6647
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 4971 . 2  |-  Ord  A
3 orduniorsuc 6638 . 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 366    = wceq 1398    e. wcel 1823   U.cuni 4235   Ord word 4866   Oncon0 4867   suc csuc 4869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873
This theorem is referenced by:  onuninsuci  6648
  Copyright terms: Public domain W3C validator