Theorem pp0ex 4781

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 4284	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 4779	. . 3 ⊢ {∅} ∈ V
3	2	pwex 4774	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2685	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128

This theorem is referenced by:  ord3ex  4782  zfpair  4831