Theorem unipw 4357

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 3938	. . . 4 |- ( x e. U. ~P A <-> E. y ( x e. y /\ y e. ~P A ) )
2		elelpwi 3701	. . . . 5 |- ( ( x e. y /\ y e. ~P A ) -> x e. A )
3	2	exlimiv 1651	. . . 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 2824	. . . . 5 |- x e. _V
6	5	snid 3740	. . . 4 |- x e. { x }
7		snelpwi 4351	. . . 4 |- ( x e. A -> { x } e. ~P A )
8		elunii 3940	. . . 4 |- ( ( x e. { x } /\ { x } e. ~P A ) -> x e. U. ~P A )
9	6, 7, 8	sylancr 418	. . 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 2235	1 |- U. ~P A = A

This proof depends on syntax axioms:   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   ~Pcpw 3688   {csn 3709   U.cuni 3935

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-uni 3936

This theorem is used by:  pwtr  4359  pwexb  4620  univ  4622  unixpss  4888  eltg4i  15156  distop  15186  distopon  15188  distps  15192  ntrss2  15222  isopn3  15226  discld  15237  txdis  15378