Theorem iinuniss 4048

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 2679	. . . 4 |- ( ( y e. A \/ A. x e. B y e. x ) -> A. x e. B ( y e. A \/ y e. x ) )
2		vex 2802	. . . . . 6 |- y e. _V
3	2	elint2 3930	. . . . 5 |- ( y e. |^| B <-> A. x e. B y e. x )
4	3	orbi2i 767	. . . 4 |- ( ( y e. A \/ y e. |^| B ) <-> ( y e. A \/ A. x e. B y e. x ) )
5		elun 3345	. . . . 5 |- ( y e. ( A u. x ) <-> ( y e. A \/ y e. x ) )
6	5	ralbii 2536	. . . 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 201	. . 3 |- ( ( y e. A \/ y e. |^| B ) -> A. x e. B y e. ( A u. x ) )
8	7	ss2abi 3296	. 2 |- { y | ( y e. A \/ y e. |^| B ) } C_ { y | A. x e. B y e. ( A u. x ) }
9		df-un 3201	. 2 |- ( A u. |^| B ) = { y | ( y e. A \/ y e. |^| B ) }
10		df-iin 3968	. 2 |- |^|_ x e. B ( A u. x ) = { y | A. x e. B y e. ( A u. x ) }
11	8, 9, 10	3sstr4i 3265	1 |- ( A u. |^| B ) C_ |^|_ x e. B ( A u. x )

Syntax hints:   \/ wo 713   e. wcel 2200   {cab 2215   A.wral 2508   u. cun 3195   C_ wss 3197   |^|cint 3923   |^|_ciin 3966

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-int 3924  df-iin 3968

This theorem is referenced by: (None)