Theorem pp0ex 5404

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 4838	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5402	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5398	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2841	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 2108  Vcvv 3488  ∅c0 4352  𝒫 cpw 4622  {csn 4648  {cpr 4650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-pw 4624  df-sn 4649  df-pr 4651

This theorem is referenced by:  ord3ex  5405  zfpair  5439