Theorem pp0ex 2777

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 2473	. 2 |- P~{(/)} = {(/), {(/)}}
2		p0ex 2776	. . 3 |- {(/)} e. V
3	2	pwex 2751	. 2 |- P~{(/)} e. V
4	1, 3	eqeltrr 1548	1 |- {(/), {(/)}} e. V

Syntax hints:   e. wcel 960  Vcvv 1814  (/)c0 2283  P~cpw 2405  {csn 2413  {cpr 2414

This theorem is referenced by:  zfpair 2783

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417

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