Theorem pp0ex 5352

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 4834	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5350	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5346	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2861	1 ⊢ {∅, {∅}} ∈ V

This proof depends on syntax axioms:   ∈ wcel 2147  Vcvv 3454  ∅c0 4281  𝒫 cpw 4560  {csn 4588  {cpr 4590

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-pw 4562  df-sn 4589  df-pr 4591

This theorem is used by:  ord3ex  5353  zfpair  5387