Theorem pwuni 4171

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 3817	. . 3 |- ( x e. A -> x C_ U. A )
2		vex 2729	. . . 4 |- x e. _V
3	2	elpw 3565	. . 3 |- ( x e. ~P U. A <-> x C_ U. A )
4	1, 3	sylibr 133	. 2 |- ( x e. A -> x e. ~P U. A )
5	4	ssriv 3146	1 |- A C_ ~P U. A

Syntax hints:   e. wcel 2136   C_ wss 3116   ~Pcpw 3559   U.cuni 3789

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561  df-uni 3790

This theorem is referenced by:  uniexb  4451  2pwuninelg  6251  istopon  12651  eltg3i  12696  mopnfss  13087