Theorem pwid 2405

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 2077	. 2 |- A (_ A
2		pwid.1	. . 3 |- A e. V
3	2	elpw 2401	. 2 |- (A e. P~A <-> A (_ A)
4	1, 3	mpbir 190	1 |- A e. P~A

Syntax hints:   e. wcel 957  Vcvv 1808   (_ wss 2044  P~cpw 2398

This theorem is referenced by:  r1ord 4638  rankpw 4667

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-in 2048  df-ss 2050  df-pw 2399

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