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Theorem pwuni 4176
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 3822 . . 3 (𝑥𝐴𝑥 𝐴)
2 vex 2733 . . . 4 𝑥 ∈ V
32elpw 3570 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥 𝐴)
41, 3sylibr 133 . 2 (𝑥𝐴𝑥 ∈ 𝒫 𝐴)
54ssriv 3151 1 𝐴 ⊆ 𝒫 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 2141  wss 3121  𝒫 cpw 3564   cuni 3794
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3566  df-uni 3795
This theorem is referenced by:  uniexb  4456  2pwuninelg  6259  istopon  12764  eltg3i  12809  mopnfss  13200
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