Theorem unipw 4250

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 3842	. . . 4 |- ( x e. U. ~P A <-> E. y ( x e. y /\ y e. ~P A ) )
2		elelpwi 3617	. . . . 5 |- ( ( x e. y /\ y e. ~P A ) -> x e. A )
3	2	exlimiv 1612	. . . 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 2766	. . . . 5 |- x e. _V
6	5	snid 3653	. . . 4 |- x e. { x }
7		snelpwi 4245	. . . 4 |- ( x e. A -> { x } e. ~P A )
8		elunii 3844	. . . 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 2193	1 |- U. ~P A = A

Syntax hints:   /\ wa 104   = wceq 1364   E.wex 1506   e. wcel 2167   ~Pcpw 3605   {csn 3622   U.cuni 3839

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-uni 3840

This theorem is referenced by:  pwtr  4252  pwexb  4509  univ  4511  unixpss  4776  eltg4i  14291  distop  14321  distopon  14323  distps  14327  ntrss2  14357  isopn3  14361  discld  14372  txdis  14513