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

Theorem onuniorsuci 6931
 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 ∈ On
Assertion
Ref Expression
onuniorsuci (𝐴 = 𝐴𝐴 = suc 𝐴)

Proof of Theorem onuniorsuci
StepHypRef Expression
1 onssi.1 . . 3 𝐴 ∈ On
21onordi 5749 . 2 Ord 𝐴
3 orduniorsuc 6922 . 2 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
42, 3ax-mp 5 1 (𝐴 = 𝐴𝐴 = suc 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 382   = wceq 1475   ∈ wcel 1977  ∪ cuni 4372  Ord word 5639  Oncon0 5640  suc csuc 5642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-suc 5646 This theorem is referenced by:  onuninsuci  6932
 Copyright terms: Public domain W3C validator