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Theorem pwuni 4086
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 3734 . . 3 (𝑥𝐴𝑥 𝐴)
2 vex 2663 . . . 4 𝑥 ∈ V
32elpw 3486 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥 𝐴)
41, 3sylibr 133 . 2 (𝑥𝐴𝑥 ∈ 𝒫 𝐴)
54ssriv 3071 1 𝐴 ⊆ 𝒫 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1465  wss 3041  𝒫 cpw 3480   cuni 3706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482  df-uni 3707
This theorem is referenced by:  uniexb  4364  2pwuninelg  6148  istopon  12107  eltg3i  12152  mopnfss  12543
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