Theorem pp0ex 5004

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 4489	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5002	. . 3 ⊢ {∅} ∈ V
3	2	pwex 4997	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2836	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 2139  Vcvv 3340  ∅c0 4058  𝒫 cpw 4302  {csn 4321  {cpr 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-pw 4304  df-sn 4322  df-pr 4324

This theorem is referenced by:  ord3ex  5005  zfpair  5053