Theorem iunpwss 2608

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

Assertion

Ref	Expression
iunpwss	|- U_x e. A P~x (_ P~U.A

Distinct variable group:   x,A

Proof of Theorem iunpwss

Step	Hyp	Ref	Expression
1		ssiun 2582	. . 3 |- (E.x e. A y (_ x -> y (_ U_x e. A x)
2		eliun 2560	. . . 4 |- (y e. U_x e. A P~x <-> E.x e. A y e. P~x)
3		visset 1804	. . . . . 6 |- y e. V
4	3	elpw 2394	. . . . 5 |- (y e. P~x <-> y (_ x)
5	4	rexbii 1660	. . . 4 |- (E.x e. A y e. P~x <-> E.x e. A y (_ x)
6	2, 5	bitr 173	. . 3 |- (y e. U_x e. A P~x <-> E.x e. A y (_ x)
7	3	elpw 2394	. . . 4 |- (y e. P~U.A <-> y (_ U.A)
8		uniiun 2591	. . . . 5 |- U.A = U_x e. A x
9	8	sseq2i 2076	. . . 4 |- (y (_ U.A <-> y (_ U_x e. A x)
10	7, 9	bitr 173	. . 3 |- (y e. P~U.A <-> y (_ U_x e. A x)
11	1, 6, 10	3imtr4 219	. 2 |- (y e. U_x e. A P~x -> y e. P~U.A)
12	11	ssriv 2059	1 |- U_x e. A P~x (_ P~U.A

Syntax hints:   e. wcel 955  E.wrex 1638   (_ wss 2037  P~cpw 2391  U.cuni 2493  U_ciun 2556

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-rex 1642  df-v 1803  df-in 2041  df-ss 2043  df-pw 2392  df-uni 2494  df-iun 2558

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