Theorem pp0ex 5304

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

Step	Hyp	Ref	Expression
1		pwpw0 4743	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5302	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5298	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2836	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 2108  Vcvv 3422  ∅c0 4253  𝒫 cpw 4530  {csn 4558  {cpr 4560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-pw 4532  df-sn 4559  df-pr 4561

This theorem is referenced by:  ord3ex  5305  zfpair  5339