Theorem pwuni 3943

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 3608	. . 3 |- ( x e. A -> x C_ U. A )
2		vex 2560	. . . 4 |- x e. _V
3	2	elpw 3365	. . 3 |- ( x e. ~P U. A <-> x C_ U. A )
4	1, 3	sylibr 137	. 2 |- ( x e. A -> x e. ~P U. A )
5	4	ssriv 2949	1 |- A C_ ~P U. A

Syntax hints:   e. wcel 1393   C_ wss 2917   ~Pcpw 3359   U.cuni 3580

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-uni 3581

This theorem is referenced by:  uniexb  4205  2pwuninelg  5898