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Theorem pwuni 3917
 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 3582 . . 3 (x Ax A)
2 vex 2538 . . . 4 x V
32elpw 3340 . . 3 (x 𝒫 Ax A)
41, 3sylibr 137 . 2 (x Ax 𝒫 A)
54ssriv 2926 1 A ⊆ 𝒫 A
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1374   ⊆ wss 2894  𝒫 cpw 3334  ∪ cuni 3554 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-ss 2908  df-pw 3336  df-uni 3555 This theorem is referenced by:  uniexb  4155  2pwuninelg  5820
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