Theorem pp0ex 4855

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 4344	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 4853	. . 3 ⊢ {∅} ∈ V
3	2	pwex 4848	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2698	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  𝒫 cpw 4158  {csn 4177  {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180

This theorem is referenced by:  ord3ex  4856  zfpair  4904