Theorem pp0ex 5347

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 4773	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5345	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5341	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2857	1 ⊢ {∅, {∅}} ∈ V

This proof depends on syntax axioms:   ∈ wcel 2144  Vcvv 3450  ∅c0 4278  𝒫 cpw 4556  {csn 4583  {cpr 4585

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-pw 4558  df-sn 4584  df-pr 4586

This theorem is used by:  ord3ex  5348  zfpair  5382