Theorem unipw 4195

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 3792	. . . 4 |- ( x e. U. ~P A <-> E. y ( x e. y /\ y e. ~P A ) )
2		elelpwi 3571	. . . . 5 |- ( ( x e. y /\ y e. ~P A ) -> x e. A )
3	2	exlimiv 1586	. . . 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 2729	. . . . 5 |- x e. _V
6	5	snid 3607	. . . 4 |- x e. { x }
7		snelpwi 4190	. . . 4 |- ( x e. A -> { x } e. ~P A )
8		elunii 3794	. . . 4 |- ( ( x e. { x } /\ { x } e. ~P A ) -> x e. U. ~P A )
9	6, 7, 8	sylancr 411	. . 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 2162	1 |- U. ~P A = A

Syntax hints:   /\ wa 103   = wceq 1343   E.wex 1480   e. wcel 2136   ~Pcpw 3559   {csn 3576   U.cuni 3789

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-uni 3790

This theorem is referenced by:  pwtr  4197  pwexb  4452  univ  4454  unixpss  4717  eltg4i  12695  distop  12725  distopon  12727  distps  12731  ntrss2  12761  isopn3  12765  discld  12776  txdis  12917