Theorem pwid 1805

Description: A set is a member of its power set. 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 1519	. 2 ⊢ A ⊆ A
2		pwid.1	. . 3 ⊢ A ∈ V
3	2	elpw 1801	. 2 ⊢ (A ∈ ℘A ↔ A ⊆ A)
4	1, 3	mpbir 165	1 ⊢ A ∈ ℘A

Syntax hints:   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ℘cpw 1798

This theorem is referenced by:  r1ord 3499  rankpw 3528

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492  df-pw 1799

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