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Theorem pwuni 4122
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 3770 . . 3 (𝑥𝐴𝑥 𝐴)
2 vex 2692 . . . 4 𝑥 ∈ V
32elpw 3519 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥 𝐴)
41, 3sylibr 133 . 2 (𝑥𝐴𝑥 ∈ 𝒫 𝐴)
54ssriv 3104 1 𝐴 ⊆ 𝒫 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1481  wss 3074  𝒫 cpw 3513   cuni 3742
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3080  df-ss 3087  df-pw 3515  df-uni 3743
This theorem is referenced by:  uniexb  4400  2pwuninelg  6186  istopon  12212  eltg3i  12257  mopnfss  12648
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