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

Theorem onpwsuc 6978
Description: The collection of ordinal numbers in the power set of an ordinal number is its successor. (Contributed by NM, 19-Oct-2004.)
Assertion
Ref Expression
onpwsuc (𝐴 ∈ On → (𝒫 𝐴 ∩ On) = suc 𝐴)

Proof of Theorem onpwsuc
StepHypRef Expression
1 eloni 5702 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordpwsuc 6977 . 2 (Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)
31, 2syl 17 1 (𝐴 ∈ On → (𝒫 𝐴 ∩ On) = suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  cin 3559  𝒫 cpw 4136  Ord word 5691  Oncon0 5692  suc csuc 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-tr 4723  df-eprel 4995  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-ord 5695  df-on 5696  df-suc 5698
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator