Theorem iunpwss 3980

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 3930	. . 3 |- ( E. x e. A y C_ x -> y C_ U_ x e. A x )
2		eliun 3892	. . . 4 |- ( y e. U_ x e. A ~P x <-> E. x e. A y e. ~P x )
3		vex 2742	. . . . . 6 |- y e. _V
4	3	elpw 3583	. . . . 5 |- ( y e. ~P x <-> y C_ x )
5	4	rexbii 2484	. . . 4 |- ( E. x e. A y e. ~P x <-> E. x e. A y C_ x )
6	2, 5	bitri 184	. . 3 |- ( y e. U_ x e. A ~P x <-> E. x e. A y C_ x )
7	3	elpw 3583	. . . 4 |- ( y e. ~P U. A <-> y C_ U. A )
8		uniiun 3942	. . . . 5 |- U. A = U_ x e. A x
9	8	sseq2i 3184	. . . 4 |- ( y C_ U. A <-> y C_ U_ x e. A x )
10	7, 9	bitri 184	. . 3 |- ( y e. ~P U. A <-> y C_ U_ x e. A x )
11	1, 6, 10	3imtr4i 201	. 2 |- ( y e. U_ x e. A ~P x -> y e. ~P U. A )
12	11	ssriv 3161	1 |- U_ x e. A ~P x C_ ~P U. A

Syntax hints:   e. wcel 2148   E.wrex 2456   C_ wss 3131   ~Pcpw 3577   U.cuni 3811   U_ciun 3888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-pw 3579  df-uni 3812  df-iun 3890

This theorem is referenced by: (None)