Theorem iinuniss 3758

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 2502	. . . 4 |- ( ( y e. A \/ A. x e. B y e. x ) -> A. x e. B ( y e. A \/ y e. x ) )
2		vex 2604	. . . . . 6 |- y e. _V
3	2	elint2 3643	. . . . 5 |- ( y e. |^| B <-> A. x e. B y e. x )
4	3	orbi2i 711	. . . 4 |- ( ( y e. A \/ y e. |^| B ) <-> ( y e. A \/ A. x e. B y e. x ) )
5		elun 3113	. . . . 5 |- ( y e. ( A u. x ) <-> ( y e. A \/ y e. x ) )
6	5	ralbii 2372	. . . 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 199	. . 3 |- ( ( y e. A \/ y e. |^| B ) -> A. x e. B y e. ( A u. x ) )
8	7	ss2abi 3066	. 2 |- { y | ( y e. A \/ y e. |^| B ) } C_ { y | A. x e. B y e. ( A u. x ) }
9		df-un 2977	. 2 |- ( A u. |^| B ) = { y | ( y e. A \/ y e. |^| B ) }
10		df-iin 3681	. 2 |- |^|_ x e. B ( A u. x ) = { y | A. x e. B y e. ( A u. x ) }
11	8, 9, 10	3sstr4i 3038	1 |- ( A u. |^| B ) C_ |^|_ x e. B ( A u. x )

Syntax hints:   \/ wo 661   e. wcel 1433   {cab 2067   A.wral 2348   u. cun 2971   C_ wss 2973   |^|cint 3636   |^|_ciin 3679

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-int 3637  df-iin 3681

This theorem is referenced by: (None)