Theorem unipw 4202

Description: A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.)

Assertion

Ref	Expression
unipw	|- U. ~P A = A

Proof of Theorem unipw

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 3799	. . . 4 |- ( x e. U. ~P A <-> E. y ( x e. y /\ y e. ~P A ) )
2		elelpwi 3578	. . . . 5 |- ( ( x e. y /\ y e. ~P A ) -> x e. A )
3	2	exlimiv 1591	. . . 4 |- ( E. y ( x e. y /\ y e. ~P A ) -> x e. A )
4	1, 3	sylbi 120	. . 3 |- ( x e. U. ~P A -> x e. A )
5		vex 2733	. . . . 5 |- x e. _V
6	5	snid 3614	. . . 4 |- x e. { x }
7		snelpwi 4197	. . . 4 |- ( x e. A -> { x } e. ~P A )
8		elunii 3801	. . . 4 |- ( ( x e. { x } /\ { x } e. ~P A ) -> x e. U. ~P A )
9	6, 7, 8	sylancr 412	. . 3 |- ( x e. A -> x e. U. ~P A )
10	4, 9	impbii 125	. 2 |- ( x e. U. ~P A <-> x e. A )
11	10	eqriv 2167	1 |- U. ~P A = A

Syntax hints:   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   ~Pcpw 3566   {csn 3583   U.cuni 3796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-uni 3797

This theorem is referenced by:  pwtr  4204  pwexb  4459  univ  4461  unixpss  4724  eltg4i  12849  distop  12879  distopon  12881  distps  12885  ntrss2  12915  isopn3  12919  discld  12930  txdis  13071