Theorem pp0ex 5341

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 5339	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5335	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2825	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 2109  Vcvv 3447  ∅c0 4296  𝒫 cpw 4563  {csn 4589  {cpr 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-pw 4565  df-sn 4590  df-pr 4592

This theorem is referenced by:  ord3ex  5342  zfpair  5376