Theorem iunpwss 3826

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 3778	. . 3 |- ( E. x e. A y C_ x -> y C_ U_ x e. A x )
2		eliun 3740	. . . 4 |- ( y e. U_ x e. A ~P x <-> E. x e. A y e. ~P x )
3		vex 2623	. . . . . 6 |- y e. _V
4	3	elpw 3439	. . . . 5 |- ( y e. ~P x <-> y C_ x )
5	4	rexbii 2386	. . . 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 3439	. . . 4 |- ( y e. ~P U. A <-> y C_ U. A )
8		uniiun 3789	. . . . 5 |- U. A = U_ x e. A x
9	8	sseq2i 3052	. . . 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 3030	1 |- U_ x e. A ~P x C_ ~P U. A

Syntax hints:   e. wcel 1439   E.wrex 2361   C_ wss 3000   ~Pcpw 3433   U.cuni 3659   U_ciun 3736

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-in 3006  df-ss 3013  df-pw 3435  df-uni 3660  df-iun 3738

This theorem is referenced by: (None)