Theorem pp0ex 5357

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 4779	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5355	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5351	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2860	1 ⊢ {∅, {∅}} ∈ V

This proof depends on syntax axioms:   ∈ wcel 2143  Vcvv 3455  ∅c0 4286  𝒫 cpw 4562  {csn 4589  {cpr 4591

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-pw 4564  df-sn 4590  df-pr 4592

This theorem is used by:  ord3ex  5358  zfpair  5392