Theorem iunpwss 2613

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 2587	. . 3 |- (E.x e. A y (_ x -> y (_ U_x e. A x)
2		eliun 2565	. . . 4 |- (y e. U_x e. A P~x <-> E.x e. A y e. P~x)
3		visset 1809	. . . . . 6 |- y e. V
4	3	elpw 2400	. . . . 5 |- (y e. P~x <-> y (_ x)
5	4	rexbii 1665	. . . 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 2400	. . . 4 |- (y e. P~U.A <-> y (_ U.A)
8		uniiun 2596	. . . . 5 |- U.A = U_x e. A x
9	8	sseq2i 2082	. . . 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 2065	1 |- U_x e. A P~x (_ P~U.A

Syntax hints:   e. wcel 956  E.wrex 1643   (_ wss 2043  P~cpw 2397  U.cuni 2498  U_ciun 2561

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-rex 1647  df-v 1808  df-in 2047  df-ss 2049  df-pw 2398  df-uni 2499  df-iun 2563

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