Theorem unipw 4235

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 3827	. . . 4 |- ( x e. U. ~P A <-> E. y ( x e. y /\ y e. ~P A ) )
2		elelpwi 3602	. . . . 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 2755	. . . . 5 |- x e. _V
6	5	snid 3638	. . . 4 |- x e. { x }
7		snelpwi 4230	. . . 4 |- ( x e. A -> { x } e. ~P A )
8		elunii 3829	. . . 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 2186	1 |- U. ~P A = A

Syntax hints:   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2160   ~Pcpw 3590   {csn 3607   U.cuni 3824

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 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-uni 3825

This theorem is referenced by:  pwtr  4237  pwexb  4492  univ  4494  unixpss  4757  eltg4i  14015  distop  14045  distopon  14047  distps  14051  ntrss2  14081  isopn3  14085  discld  14096  txdis  14237