Theorem pp0ex 5358

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 4783	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5356	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5352	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2866	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 2149  Vcvv 3463  ∅c0 4294  𝒫 cpw 4567  {csn 4594  {cpr 4596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-pw 4569  df-sn 4595  df-pr 4597

This theorem is referenced by:  ord3ex  5359  zfpair  5393