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Theorem pp0ex 5386
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 4813 . 2 𝒫 {∅} = {∅, {∅}}
2 p0ex 5384 . . 3 {∅} ∈ V
32pwex 5380 . 2 𝒫 {∅} ∈ V
41, 3eqeltrri 2838 1 {∅, {∅}} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  Vcvv 3480  c0 4333  𝒫 cpw 4600  {csn 4626  {cpr 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-pw 4602  df-sn 4627  df-pr 4629
This theorem is referenced by:  ord3ex  5387  zfpair  5421
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