Theorem pp0ex 5366

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 4793	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5364	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5360	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2830	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 2107  Vcvv 3463  ∅c0 4313  𝒫 cpw 4580  {csn 4606  {cpr 4608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-pw 4582  df-sn 4607  df-pr 4609

This theorem is referenced by:  ord3ex  5367  zfpair  5401