Theorem pp0ex 5309

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 4746	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5307	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5303	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2836	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 2106  Vcvv 3432  ∅c0 4256  𝒫 cpw 4533  {csn 4561  {cpr 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-pw 4535  df-sn 4562  df-pr 4564

This theorem is referenced by:  ord3ex  5310  zfpair  5344