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Theorem pp0ex 5255
 Description: The power set of the power set of the empty set (the ordinal 2) is a set. (Contributed by NM, 24-Jun-1993.)
Assertion
Ref Expression
pp0ex {∅, {∅}} ∈ V

Proof of Theorem pp0ex
StepHypRef Expression
1 pwpw0 4709 . 2 𝒫 {∅} = {∅, {∅}}
2 p0ex 5253 . . 3 {∅} ∈ V
32pwex 5249 . 2 𝒫 {∅} ∈ V
41, 3eqeltrri 2890 1 {∅, {∅}} ∈ V
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2112  Vcvv 3444  ∅c0 4246  𝒫 cpw 4500  {csn 4528  {cpr 4530 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-pw 4502  df-sn 4529  df-pr 4531 This theorem is referenced by:  ord3ex  5256  zfpair  5290
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