Theorem pwuni 4275

Description: A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.)

Assertion

Ref	Expression
pwuni	|- A C_ ~P U. A

Proof of Theorem pwuni

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elssuni 3915	. . 3 |- ( x e. A -> x C_ U. A )
2		vex 2802	. . . 4 |- x e. _V
3	2	elpw 3655	. . 3 |- ( x e. ~P U. A <-> x C_ U. A )
4	1, 3	sylibr 134	. 2 |- ( x e. A -> x e. ~P U. A )
5	4	ssriv 3228	1 |- A C_ ~P U. A

Syntax hints:   e. wcel 2200   C_ wss 3197   ~Pcpw 3649   U.cuni 3887

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651  df-uni 3888

This theorem is referenced by:  uniexb  4563  2pwuninelg  6427  istopon  14681  eltg3i  14724  mopnfss  15115