Theorem pp0ex 2771

Description: The power set of the power set of the empty set is a set.

Assertion

Ref	Expression
pp0ex	|- {(/), {(/)}} e. V

Proof of Theorem pp0ex

Step	Hyp	Ref	Expression
1		pwpw0 2469	. 2 |- P~{(/)} = {(/), {(/)}}
2		p0ex 2770	. . 3 |- {(/)} e. V
3	2	pwex 2745	. 2 |- P~{(/)} e. V
4	1, 3	eqeltrr 1545	1 |- {(/), {(/)}} e. V

Syntax hints:   e. wcel 958  Vcvv 1811  (/)c0 2280  P~cpw 2401  {csn 2409  {cpr 2410

This theorem is referenced by:  zfpair 2777

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413

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