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Theorem unipw 3927
 Description: A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.)
Assertion
Ref Expression
unipw 𝒫 A = A

Proof of Theorem unipw
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni 3557 . . . 4 (x 𝒫 Ay(x y y 𝒫 A))
2 elelpwi 3345 . . . . 5 ((x y y 𝒫 A) → x A)
32exlimiv 1471 . . . 4 (y(x y y 𝒫 A) → x A)
41, 3sylbi 114 . . 3 (x 𝒫 Ax A)
5 vex 2538 . . . . 5 x V
65snid 3377 . . . 4 x {x}
7 snelpwi 3922 . . . 4 (x A → {x} 𝒫 A)
8 elunii 3559 . . . 4 ((x {x} {x} 𝒫 A) → x 𝒫 A)
96, 7, 8sylancr 395 . . 3 (x Ax 𝒫 A)
104, 9impbii 117 . 2 (x 𝒫 Ax A)
1110eqriv 2019 1 𝒫 A = A
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1228  ∃wex 1362   ∈ wcel 1374  𝒫 cpw 3334  {csn 3350  ∪ 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-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901 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-sn 3356  df-uni 3555 This theorem is referenced by:  pwtr  3929  pwexb  4156  univ  4157  unixpss  4378
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