Theorem pp0ex 5384

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 4816	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5382	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5378	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2829	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 2105  Vcvv 3473  ∅c0 4322  𝒫 cpw 4602  {csn 4628  {cpr 4630

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-pw 4604  df-sn 4629  df-pr 4631

This theorem is referenced by:  ord3ex  5385  zfpair  5419