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

Step	Hyp	Ref	Expression
1		pwpw0 4813	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5384	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5380	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2838	1 ⊢ {∅, {∅}} ∈ V

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