Theorem pwuni 3934

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 3599	. . 3 |- (x e. A -> x C_ U.A)
2		vex 2554	. . . 4 |- x e. _V
3	2	elpw 3357	. . 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 2943	1 |- A C_ ~P U.A

Syntax hints:   e. wcel 1390   C_ wss 2911   ~Pcpw 3351   U.cuni 3571

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-uni 3572

This theorem is referenced by:  uniexb  4171  2pwuninelg  5839