Theorem iunpwss 3899

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 3850	. . 3 |- ( E. x e. A y C_ x -> y C_ U_ x e. A x )
2		eliun 3812	. . . 4 |- ( y e. U_ x e. A ~P x <-> E. x e. A y e. ~P x )
3		vex 2684	. . . . . 6 |- y e. _V
4	3	elpw 3511	. . . . 5 |- ( y e. ~P x <-> y C_ x )
5	4	rexbii 2440	. . . 4 |- ( E. x e. A y e. ~P x <-> E. x e. A y C_ x )
6	2, 5	bitri 183	. . 3 |- ( y e. U_ x e. A ~P x <-> E. x e. A y C_ x )
7	3	elpw 3511	. . . 4 |- ( y e. ~P U. A <-> y C_ U. A )
8		uniiun 3861	. . . . 5 |- U. A = U_ x e. A x
9	8	sseq2i 3119	. . . 4 |- ( y C_ U. A <-> y C_ U_ x e. A x )
10	7, 9	bitri 183	. . 3 |- ( y e. ~P U. A <-> y C_ U_ x e. A x )
11	1, 6, 10	3imtr4i 200	. 2 |- ( y e. U_ x e. A ~P x -> y e. ~P U. A )
12	11	ssriv 3096	1 |- U_ x e. A ~P x C_ ~P U. A

Syntax hints:   e. wcel 1480   E.wrex 2415   C_ wss 3066   ~Pcpw 3505   U.cuni 3731   U_ciun 3808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-pw 3507  df-uni 3732  df-iun 3810

This theorem is referenced by: (None)