Theorem pp0ex 5264

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 4712	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5262	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5258	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2828	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 2112  Vcvv 3398  ∅c0 4223  𝒫 cpw 4499  {csn 4527  {cpr 4529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-pw 4501  df-sn 4528  df-pr 4530

This theorem is referenced by:  ord3ex  5265  zfpair  5299