Theorem pwv 1884

Description: The power class of the universe is the universe. Exercise 4.12(d) of [Mendelson] p. 235.

Assertion

Ref	Expression
pwv	⊢ ℘V = V

Proof of Theorem pwv

Step	Hyp	Ref	Expression
1		ssv 1520	. . . 4 ⊢ x ⊆ V
2		visset 1350	. . . . 5 ⊢ x ∈ V
3	2	elpw 1801	. . . 4 ⊢ (x ∈ ℘V ↔ x ⊆ V)
4	1, 3	mpbir 165	. . 3 ⊢ x ∈ ℘V
5	4, 2	2th 540	. 2 ⊢ (x ∈ ℘V ↔ x ∈ V)
6	5	cleqri 1101	1 ⊢ ℘V = V

Syntax hints:   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ℘cpw 1798

This theorem is referenced by:  univ 1964  ncanth 2946

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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