Theorem pp0ex 5343

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 4771	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5341	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5337	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2859	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 2142  Vcvv 3454  ∅c0 4285  𝒫 cpw 4555  {csn 4582  {cpr 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-pw 4557  df-sn 4583  df-pr 4585

This theorem is referenced by:  ord3ex  5344  zfpair  5378