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Theorem unipw 3981
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 3611 . . . 4 (𝑥 𝒫 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦 ∈ 𝒫 𝐴))
2 elelpwi 3398 . . . . 5 ((𝑥𝑦𝑦 ∈ 𝒫 𝐴) → 𝑥𝐴)
32exlimiv 1505 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ 𝒫 𝐴) → 𝑥𝐴)
41, 3sylbi 118 . . 3 (𝑥 𝒫 𝐴𝑥𝐴)
5 vex 2577 . . . . 5 𝑥 ∈ V
65snid 3430 . . . 4 𝑥 ∈ {𝑥}
7 snelpwi 3976 . . . 4 (𝑥𝐴 → {𝑥} ∈ 𝒫 𝐴)
8 elunii 3613 . . . 4 ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 𝒫 𝐴)
96, 7, 8sylancr 399 . . 3 (𝑥𝐴𝑥 𝒫 𝐴)
104, 9impbii 121 . 2 (𝑥 𝒫 𝐴𝑥𝐴)
1110eqriv 2053 1 𝒫 𝐴 = 𝐴
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wex 1397  wcel 1409  𝒫 cpw 3387  {csn 3403   cuni 3608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-uni 3609
This theorem is referenced by:  pwtr  3983  pwexb  4234  univ  4235  unixpss  4479
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