Theorem iinuniss 3969

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 2625	. . . 4 |- ( ( y e. A \/ A. x e. B y e. x ) -> A. x e. B ( y e. A \/ y e. x ) )
2		vex 2740	. . . . . 6 |- y e. _V
3	2	elint2 3851	. . . . 5 |- ( y e. |^| B <-> A. x e. B y e. x )
4	3	orbi2i 762	. . . 4 |- ( ( y e. A \/ y e. |^| B ) <-> ( y e. A \/ A. x e. B y e. x ) )
5		elun 3276	. . . . 5 |- ( y e. ( A u. x ) <-> ( y e. A \/ y e. x ) )
6	5	ralbii 2483	. . . 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 3227	. 2 |- { y | ( y e. A \/ y e. |^| B ) } C_ { y | A. x e. B y e. ( A u. x ) }
9		df-un 3133	. 2 |- ( A u. |^| B ) = { y | ( y e. A \/ y e. |^| B ) }
10		df-iin 3889	. 2 |- |^|_ x e. B ( A u. x ) = { y | A. x e. B y e. ( A u. x ) }
11	8, 9, 10	3sstr4i 3196	1 |- ( A u. |^| B ) C_ |^|_ x e. B ( A u. x )

Syntax hints:   \/ wo 708   e. wcel 2148   {cab 2163   A.wral 2455   u. cun 3127   C_ wss 3129   |^|cint 3844   |^|_ciin 3887

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-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-int 3845  df-iin 3889

This theorem is referenced by: (None)