Theorem pp0ex 2766

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 2465	. 2 |- P~{(/)} = {(/), {(/)}}
2		p0ex 2765	. . 3 |- {(/)} e. V
3	2	pwex 2740	. 2 |- P~{(/)} e. V
4	1, 3	eqeltrr 1542	1 |- {(/), {(/)}} e. V

Syntax hints:   e. wcel 956  Vcvv 1807  (/)c0 2276  P~cpw 2397  {csn 2405  {cpr 2406

This theorem is referenced by:  zfpair 2772

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409

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