Theorem pp0ex 5324

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 4757	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5322	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5318	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2834	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 2114  Vcvv 3430  ∅c0 4274  𝒫 cpw 4542  {csn 4568  {cpr 4570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-pw 4544  df-sn 4569  df-pr 4571

This theorem is referenced by:  ord3ex  5325  zfpair  5359