Theorem pwid 2418

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

Hypothesis

Ref	Expression
pwid.1	⊢ A ∈ V

Assertion

Ref	Expression
pwid	⊢ A ∈ ℘A

Proof of Theorem pwid

Step	Hyp	Ref	Expression
1		ssid 2089	. 2 ⊢ A ⊆ A
2		pwid.1	. . 3 ⊢ A ∈ V
3	2	elpw 2414	. 2 ⊢ (A ∈ ℘A ↔ A ⊆ A)
4	1, 3	mpbir 190	1 ⊢ A ∈ ℘A

Syntax hints:   ∈ wcel 962  Vcvv 1818   ⊆ wss 2056  ℘cpw 2411

This theorem is referenced by:  r1ord 4667  rankpw 4696

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 985  df-sb 1178  df-clab 1471  df-cleq 1476  df-clel 1479  df-v 1819  df-in 2060  df-ss 2062  df-pw 2412

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