Theorem pp0ex 5139

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 4620	. 2 ⊢ 𝒫 {∅} = {∅, {∅}}
2		p0ex 5137	. . 3 ⊢ {∅} ∈ V
3	2	pwex 5134	. 2 ⊢ 𝒫 {∅} ∈ V
4	1, 3	eqeltrri 2863	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 2050  Vcvv 3415  ∅c0 4178  𝒫 cpw 4422  {csn 4441  {cpr 4443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-v 3417  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-pw 4424  df-sn 4442  df-pr 4444

This theorem is referenced by:  ord3ex  5140  zfpair  5178