Theorem iunpwss 4618

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 4562	. . 3 |- ( E. x e. A y C_ x -> y C_ U_ x e. A x )
2		eliun 4524	. . . 4 |- ( y e. U_ x e. A ~P x <-> E. x e. A y e. ~P x )
3		selpw 4165	. . . . 5 |- ( y e. ~P x <-> y C_ x )
4	3	rexbii 3041	. . . 4 |- ( E. x e. A y e. ~P x <-> E. x e. A y C_ x )
5	2, 4	bitri 264	. . 3 |- ( y e. U_ x e. A ~P x <-> E. x e. A y C_ x )
6		selpw 4165	. . . 4 |- ( y e. ~P U. A <-> y C_ U. A )
7		uniiun 4573	. . . . 5 |- U. A = U_ x e. A x
8	7	sseq2i 3630	. . . 4 |- ( y C_ U. A <-> y C_ U_ x e. A x )
9	6, 8	bitri 264	. . 3 |- ( y e. ~P U. A <-> y C_ U_ x e. A x )
10	1, 5, 9	3imtr4i 281	. 2 |- ( y e. U_ x e. A ~P x -> y e. ~P U. A )
11	10	ssriv 3607	1 |- U_ x e. A ~P x C_ ~P U. A

Syntax hints:   e. wcel 1990   E.wrex 2913   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   U_ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-iun 4522

This theorem is referenced by: (None)