Theorem pp0ex 5315

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 4744	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5313	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5309	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2836	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 2119  Vcvv 3431  ∅c0 4261  𝒫 cpw 4529  {csn 4555  {cpr 4557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-pw 4531  df-sn 4556  df-pr 4558

This theorem is referenced by:  ord3ex  5316  zfpair  5350