Theorem pp0ex 5392

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 4818	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5390	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5386	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2836	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 2106  Vcvv 3478  ∅c0 4339  𝒫 cpw 4605  {csn 4631  {cpr 4633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-pw 4607  df-sn 4632  df-pr 4634

This theorem is referenced by:  ord3ex  5393  zfpair  5427