Theorem unipw 4246

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 3838	. . . 4 |- ( x e. U. ~P A <-> E. y ( x e. y /\ y e. ~P A ) )
2		elelpwi 3613	. . . . 5 |- ( ( x e. y /\ y e. ~P A ) -> x e. A )
3	2	exlimiv 1609	. . . 4 |- ( E. y ( x e. y /\ y e. ~P A ) -> x e. A )
4	1, 3	sylbi 121	. . 3 |- ( x e. U. ~P A -> x e. A )
5		vex 2763	. . . . 5 |- x e. _V
6	5	snid 3649	. . . 4 |- x e. { x }
7		snelpwi 4241	. . . 4 |- ( x e. A -> { x } e. ~P A )
8		elunii 3840	. . . 4 |- ( ( x e. { x } /\ { x } e. ~P A ) -> x e. U. ~P A )
9	6, 7, 8	sylancr 414	. . 3 |- ( x e. A -> x e. U. ~P A )
10	4, 9	impbii 126	. 2 |- ( x e. U. ~P A <-> x e. A )
11	10	eqriv 2190	1 |- U. ~P A = A

Syntax hints:   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2164   ~Pcpw 3601   {csn 3618   U.cuni 3835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-uni 3836

This theorem is referenced by:  pwtr  4248  pwexb  4505  univ  4507  unixpss  4772  eltg4i  14223  distop  14253  distopon  14255  distps  14259  ntrss2  14289  isopn3  14293  discld  14304  txdis  14445