Theorem pp0ex 5344

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 4780	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5342	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5338	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2826	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 2109  Vcvv 3450  ∅c0 4299  𝒫 cpw 4566  {csn 4592  {cpr 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-pw 4568  df-sn 4593  df-pr 4595

This theorem is referenced by:  ord3ex  5345  zfpair  5379