Theorem pwid 2398

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 2070	. 2 |- A (_ A
2		pwid.1	. . 3 |- A e. V
3	2	elpw 2394	. 2 |- (A e. P~A <-> A (_ A)
4	1, 3	mpbir 190	1 |- A e. P~A

Syntax hints:   e. wcel 955  Vcvv 1802   (_ wss 2037  P~cpw 2391

This theorem is referenced by:  r1ord 4627  rankpw 4656

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-in 2041  df-ss 2043  df-pw 2392

Copyright terms: Public domain