Theorem pp0ex 5278

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 4739	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5276	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5273	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2910	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 2110  Vcvv 3494  ∅c0 4290  𝒫 cpw 4538  {csn 4560  {cpr 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-pw 4540  df-sn 4561  df-pr 4563

This theorem is referenced by:  ord3ex  5279  zfpair  5313