Theorem pp0ex 5355

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 4777	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5353	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5349	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2859	1 ⊢ {∅, {∅}} ∈ V

This proof depends on syntax axioms:   ∈ wcel 2145  Vcvv 3453  ∅c0 4282  𝒫 cpw 4560  {csn 4587  {cpr 4589

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-pw 4562  df-sn 4588  df-pr 4590

This theorem is used by:  ord3ex  5356  zfpair  5390