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Theorem onpwsuc 7808
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 6374 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordpwsuc 7807 . 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 1540  wcel 2105  cin 3947  𝒫 cpw 4602  Ord word 6363  Oncon0 6364  suc csuc 6366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368  df-suc 6370
This theorem is referenced by: (None)
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