Theorem iunpwss 3337

Description: Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33.

Assertion

Ref	Expression
iunpwss	|- U_x e. A ~Px C_ ~PU.A

Distinct variable group:   x,A

Proof of Theorem iunpwss

Step	Hyp	Ref	Expression
1		ssiun 3293	. . 3 |- (E.x e. A y C_ x -> y C_ U_x e. A x)
2		eliun 3259	. . . 4 |- (y e. U_x e. A ~Px <-> E.x e. A y e. ~Px)
3		visset 2295	. . . . . 6 |- y e. _V
4	3	elpw 3037	. . . . 5 |- (y e. ~Px <-> y C_ x)
5	4	rexbii 2128	. . . 4 |- (E.x e. A y e. ~Px <-> E.x e. A y C_ x)
6	2, 5	bitri 190	. . 3 |- (y e. U_x e. A ~Px <-> E.x e. A y C_ x)
7	3	elpw 3037	. . . 4 |- (y e. ~PU.A <-> y C_ U.A)
8		uniiun 3306	. . . . 5 |- U.A = U_x e. A x
9	8	sseq2i 2642	. . . 4 |- (y C_ U.A <-> y C_ U_x e. A x)
10	7, 9	bitri 190	. . 3 |- (y e. ~PU.A <-> y C_ U_x e. A x)
11	1, 6, 10	3imtr4i 236	. 2 |- (y e. U_x e. A ~Px -> y e. ~PU.A)
12	11	ssriv 2621	1 |- U_x e. A ~Px C_ ~PU.A

Syntax hints:   e. wcel 1300  E.wrex 2106   C_ wss 2593  ~Pcpw 3032  U.cuni 3177  U_ciun 3255

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-in 2603  df-ss 2605  df-pw 3035  df-uni 3178  df-iun 3257

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