Theorem unipw 4139

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 3739	. . . 4 |- ( x e. U. ~P A <-> E. y ( x e. y /\ y e. ~P A ) )
2		elelpwi 3522	. . . . 5 |- ( ( x e. y /\ y e. ~P A ) -> x e. A )
3	2	exlimiv 1577	. . . 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 2689	. . . . 5 |- x e. _V
6	5	snid 3556	. . . 4 |- x e. { x }
7		snelpwi 4134	. . . 4 |- ( x e. A -> { x } e. ~P A )
8		elunii 3741	. . . 4 |- ( ( x e. { x } /\ { x } e. ~P A ) -> x e. U. ~P A )
9	6, 7, 8	sylancr 410	. . 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 2136	1 |- U. ~P A = A

Syntax hints:   /\ wa 103   = wceq 1331   E.wex 1468   e. wcel 1480   ~Pcpw 3510   {csn 3527   U.cuni 3736

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-uni 3737

This theorem is referenced by:  pwtr  4141  pwexb  4395  univ  4397  unixpss  4652  eltg4i  12224  distop  12254  distopon  12256  distps  12260  ntrss2  12290  isopn3  12294  discld  12305  txdis  12446