Theorem uniinn0 29366

Description: Sufficient and necessary condition for a union to intersect with a given set. (Contributed by Thierry Arnoux, 27-Jan-2020.)

Assertion

Ref	Expression
uniinn0	|- ( ( U. A i^i B ) =/= (/) <-> E. x e. A ( x i^i B ) =/= (/) )

Distinct variable groups:   x, A   x, B

Proof of Theorem uniinn0

Step	Hyp	Ref	Expression
1		nne 2798	. . . 4 |- ( -. ( x i^i B ) =/= (/) <-> ( x i^i B ) = (/) )
2	1	ralbii 2980	. . 3 |- ( A. x e. A -. ( x i^i B ) =/= (/) <-> A. x e. A ( x i^i B ) = (/) )
3		ralnex 2992	. . 3 |- ( A. x e. A -. ( x i^i B ) =/= (/) <-> -. E. x e. A ( x i^i B ) =/= (/) )
4		unissb 4469	. . . 4 |- ( U. A C_ ( _V \ B ) <-> A. x e. A x C_ ( _V \ B ) )
5		disj2 4024	. . . 4 |- ( ( U. A i^i B ) = (/) <-> U. A C_ ( _V \ B ) )
6		disj2 4024	. . . . 5 |- ( ( x i^i B ) = (/) <-> x C_ ( _V \ B ) )
7	6	ralbii 2980	. . . 4 |- ( A. x e. A ( x i^i B ) = (/) <-> A. x e. A x C_ ( _V \ B ) )
8	4, 5, 7	3bitr4ri 293	. . 3 |- ( A. x e. A ( x i^i B ) = (/) <-> ( U. A i^i B ) = (/) )
9	2, 3, 8	3bitr3i 290	. 2 |- ( -. E. x e. A ( x i^i B ) =/= (/) <-> ( U. A i^i B ) = (/) )
10	9	necon1abii 2842	1 |- ( ( U. A i^i B ) =/= (/) <-> E. x e. A ( x i^i B ) =/= (/) )

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   U.cuni 4436

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-uni 4437

This theorem is referenced by:  locfinreflem  29907