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Theorem pwuni 4945
Description: A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.)
Assertion
Ref Expression
pwuni 𝐴 ⊆ 𝒫 𝐴

Proof of Theorem pwuni
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elssuni 4937 . . 3 (𝑥𝐴𝑥 𝐴)
2 velpw 4602 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥 𝐴)
31, 2sylibr 233 . 2 (𝑥𝐴𝑥 ∈ 𝒫 𝐴)
43ssriv 3982 1 𝐴 ⊆ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2099  wss 3946  𝒫 cpw 4597   cuni 4905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-ss 3963  df-pw 4599  df-uni 4906
This theorem is referenced by:  uniexr  7763  fipwuni  9462  uniwf  9855  rankuni  9899  rankc2  9907  rankxplim  9915  fin23lem17  10372  axcclem  10491  grurn  10835  istopon  22902  eltg3i  22952  cmpfi  23400  hmphdis  23788  ptcmpfi  23805  fbssfi  23829  mopnfss  24437  pliguhgr  30416  shsspwh  31176  circtopn  33665  hasheuni  33931  issgon  33969  sigaclci  33978  sigagenval  33986  dmsigagen  33990  imambfm  34109  bj-unirel  36771  salgenval  45978  salgenn0  45988  caragensspw  46166
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