Theorem pwid 2408

Description: A set is a member of its power class. Theorem 87 of [Suppes] p. 47.

Hypothesis

Ref	Expression
pwid.1	|- A e. V

Assertion

Ref	Expression
pwid	|- A e. P~A

Proof of Theorem pwid

Step	Hyp	Ref	Expression
1		ssid 2080	. 2 |- A (_ A
2		pwid.1	. . 3 |- A e. V
3	2	elpw 2404	. 2 |- (A e. P~A <-> A (_ A)
4	1, 3	mpbir 190	1 |- A e. P~A

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

This theorem is referenced by:  r1ord 4655  rankpw 4684

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

Copyright terms: Public domain