Theorem pp0ex 5329

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 4767	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5327	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5323	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2831	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 2113  Vcvv 3438  ∅c0 4283  𝒫 cpw 4552  {csn 4578  {cpr 4580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-pw 4554  df-sn 4579  df-pr 4581

This theorem is referenced by:  ord3ex  5330  zfpair  5364