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Theorem pp0ex 5004
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 4489 . 2 𝒫 {∅} = {∅, {∅}}
2 p0ex 5002 . . 3 {∅} ∈ V
32pwex 4997 . 2 𝒫 {∅} ∈ V
41, 3eqeltrri 2836 1 {∅, {∅}} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2139  Vcvv 3340  c0 4058  𝒫 cpw 4302  {csn 4321  {cpr 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-pw 4304  df-sn 4322  df-pr 4324
This theorem is referenced by:  ord3ex  5005  zfpair  5053
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