Theorem iunpwss 3743

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. (Contributed by NM, 25-Nov-2003.)

Assertion

Ref	Expression
iunpwss	|- U_ x e. A ~P x C_ ~P U. A

Distinct variable group:   x, A

Proof of Theorem iunpwss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssiun 3699	. . 3 |- ( E. x e. A y C_ x -> y C_ U_ x e. A x )
2		eliun 3661	. . . 4 |- ( y e. U_ x e. A ~P x <-> E. x e. A y e. ~P x )
3		vex 2560	. . . . . 6 |- y e. _V
4	3	elpw 3365	. . . . 5 |- ( y e. ~P x <-> y C_ x )
5	4	rexbii 2331	. . . 4 |- ( E. x e. A y e. ~P x <-> E. x e. A y C_ x )
6	2, 5	bitri 173	. . 3 |- ( y e. U_ x e. A ~P x <-> E. x e. A y C_ x )
7	3	elpw 3365	. . . 4 |- ( y e. ~P U. A <-> y C_ U. A )
8		uniiun 3710	. . . . 5 |- U. A = U_ x e. A x
9	8	sseq2i 2970	. . . 4 |- ( y C_ U. A <-> y C_ U_ x e. A x )
10	7, 9	bitri 173	. . 3 |- ( y e. ~P U. A <-> y C_ U_ x e. A x )
11	1, 6, 10	3imtr4i 190	. 2 |- ( y e. U_ x e. A ~P x -> y e. ~P U. A )
12	11	ssriv 2949	1 |- U_ x e. A ~P x C_ ~P U. A

Syntax hints:   e. wcel 1393   E.wrex 2307   C_ wss 2917   ~Pcpw 3359   U.cuni 3580   U_ciun 3657

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-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-uni 3581  df-iun 3659

This theorem is referenced by: (None)