Theorem pwv 2502

Description: The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235.

Assertion

Ref	Expression
pwv	|- P~V = V

Proof of Theorem pwv

Step	Hyp	Ref	Expression
1		ssv 2081	. . . 4 |- x (_ V
2		visset 1813	. . . . 5 |- x e. V
3	2	elpw 2404	. . . 4 |- (x e. P~V <-> x (_ V)
4	1, 3	mpbir 190	. . 3 |- x e. P~V
5	4, 2	2th 718	. 2 |- (x e. P~V <-> x e. V)
6	5	eqriv 1474	1 |- P~V = V

Syntax hints:   = wceq 956   e. wcel 958  Vcvv 1811   (_ wss 2047  P~cpw 2401

This theorem is referenced by:  univ 2909  ncanth 3908

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-12 968  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

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-ss 2053  df-pw 2402

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