Theorem iinssiun 29377

Description: An indexed intersection is a subset of the corresponding indexed union. (Contributed by Thierry Arnoux, 31-Dec-2021.)

Assertion

Ref	Expression
iinssiun	|- ( A =/= (/) -> |^|_ x e. A B C_ U_ x e. A B )

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem iinssiun

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.2z 4060	. . . 4 |- ( ( A =/= (/) /\ A. x e. A y e. B ) -> E. x e. A y e. B )
2	1	ex 450	. . 3 |- ( A =/= (/) -> ( A. x e. A y e. B -> E. x e. A y e. B ) )
3		vex 3203	. . . 4 |- y e. _V
4		eliin 4525	. . . 4 |- ( y e. _V -> ( y e. |^|_ x e. A B <-> A. x e. A y e. B ) )
5	3, 4	ax-mp 5	. . 3 |- ( y e. |^|_ x e. A B <-> A. x e. A y e. B )
6		eliun 4524	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
7	2, 5, 6	3imtr4g 285	. 2 |- ( A =/= (/) -> ( y e. |^|_ x e. A B -> y e. U_ x e. A B ) )
8	7	ssrdv 3609	1 |- ( A =/= (/) -> |^|_ x e. A B C_ U_ x e. A B )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   U_ciun 4520   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-iun 4522  df-iin 4523

This theorem is referenced by: (None)