Theorem iinuniss 3895

Description: A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33 but with equality changed to subset. (Contributed by Jim Kingdon, 19-Aug-2018.)

Assertion

Ref	Expression
iinuniss	|- ( A u. |^| B ) C_ |^|_ x e. B ( A u. x )

Distinct variable groups:   x, A   x, B

Proof of Theorem iinuniss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.32vr 2579	. . . 4 |- ( ( y e. A \/ A. x e. B y e. x ) -> A. x e. B ( y e. A \/ y e. x ) )
2		vex 2689	. . . . . 6 |- y e. _V
3	2	elint2 3778	. . . . 5 |- ( y e. |^| B <-> A. x e. B y e. x )
4	3	orbi2i 751	. . . 4 |- ( ( y e. A \/ y e. |^| B ) <-> ( y e. A \/ A. x e. B y e. x ) )
5		elun 3217	. . . . 5 |- ( y e. ( A u. x ) <-> ( y e. A \/ y e. x ) )
6	5	ralbii 2441	. . . 4 |- ( A. x e. B y e. ( A u. x ) <-> A. x e. B ( y e. A \/ y e. x ) )
7	1, 4, 6	3imtr4i 200	. . 3 |- ( ( y e. A \/ y e. |^| B ) -> A. x e. B y e. ( A u. x ) )
8	7	ss2abi 3169	. 2 |- { y | ( y e. A \/ y e. |^| B ) } C_ { y | A. x e. B y e. ( A u. x ) }
9		df-un 3075	. 2 |- ( A u. |^| B ) = { y | ( y e. A \/ y e. |^| B ) }
10		df-iin 3816	. 2 |- |^|_ x e. B ( A u. x ) = { y | A. x e. B y e. ( A u. x ) }
11	8, 9, 10	3sstr4i 3138	1 |- ( A u. |^| B ) C_ |^|_ x e. B ( A u. x )

Syntax hints:   \/ wo 697   e. wcel 1480   {cab 2125   A.wral 2416   u. cun 3069   C_ wss 3071   |^|cint 3771   |^|_ciin 3814

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 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-int 3772  df-iin 3816

This theorem is referenced by: (None)