Theorem unipw 4097

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 3703	. . . 4 |- ( x e. U. ~P A <-> E. y ( x e. y /\ y e. ~P A ) )
2		elelpwi 3486	. . . . 5 |- ( ( x e. y /\ y e. ~P A ) -> x e. A )
3	2	exlimiv 1558	. . . 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 2658	. . . . 5 |- x e. _V
6	5	snid 3520	. . . 4 |- x e. { x }
7		snelpwi 4092	. . . 4 |- ( x e. A -> { x } e. ~P A )
8		elunii 3705	. . . 4 |- ( ( x e. { x } /\ { x } e. ~P A ) -> x e. U. ~P A )
9	6, 7, 8	sylancr 408	. . 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 2110	1 |- U. ~P A = A

Syntax hints:   /\ wa 103   = wceq 1312   E.wex 1449   e. wcel 1461   ~Pcpw 3474   {csn 3491   U.cuni 3700

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-uni 3701

This theorem is referenced by:  pwtr  4099  pwexb  4353  univ  4355  unixpss  4610  eltg4i  12061  distop  12091  distopon  12093  distps  12097  ntrss2  12127  isopn3  12131  discld  12142  txdis  12282