Theorem iunpwss 4099

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 4049	. . 3 |- ( E. x e. A y C_ x -> y C_ U_ x e. A x )
2		eliun 4011	. . . 4 |- ( y e. U_ x e. A ~P x <-> E. x e. A y e. ~P x )
3		vex 2824	. . . . . 6 |- y e. _V
4	3	elpw 3691	. . . . 5 |- ( y e. ~P x <-> y C_ x )
5	4	rexbii 2557	. . . 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 3691	. . . 4 |- ( y e. ~P U. A <-> y C_ U. A )
8		uniiun 4061	. . . . 5 |- U. A = U_ x e. A x
9	8	sseq2i 3275	. . . 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 3252	1 |- U_ x e. A ~P x C_ ~P U. A

Syntax hints:   e. wcel 2209   E.wrex 2529   C_ wss 3220   ~Pcpw 3685   U.cuni 3930   U_ciun 4007

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687  df-uni 3931  df-iun 4009

This theorem is referenced by: (None)