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Theorem uniinn0 28163
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
StepHypRef Expression
1 nne 2628 . . . 4  |-  ( -.  ( x  i^i  B
)  =/=  (/)  <->  ( x  i^i  B )  =  (/) )
21ralbii 2819 . . 3  |-  ( A. x  e.  A  -.  ( x  i^i  B )  =/=  (/)  <->  A. x  e.  A  ( x  i^i  B )  =  (/) )
3 ralnex 2834 . . 3  |-  ( A. x  e.  A  -.  ( x  i^i  B )  =/=  (/)  <->  -.  E. x  e.  A  ( x  i^i  B )  =/=  (/) )
4 unissb 4229 . . . 4  |-  ( U. A  C_  ( _V  \  B )  <->  A. x  e.  A  x  C_  ( _V  \  B ) )
5 disj2 3812 . . . 4  |-  ( ( U. A  i^i  B
)  =  (/)  <->  U. A  C_  ( _V  \  B ) )
6 disj2 3812 . . . . 5  |-  ( ( x  i^i  B )  =  (/)  <->  x  C_  ( _V 
\  B ) )
76ralbii 2819 . . . 4  |-  ( A. x  e.  A  (
x  i^i  B )  =  (/)  <->  A. x  e.  A  x  C_  ( _V  \  B ) )
84, 5, 73bitr4ri 282 . . 3  |-  ( A. x  e.  A  (
x  i^i  B )  =  (/)  <->  ( U. A  i^i  B )  =  (/) )
92, 3, 83bitr3i 279 . 2  |-  ( -. 
E. x  e.  A  ( x  i^i  B )  =/=  (/)  <->  ( U. A  i^i  B )  =  (/) )
109necon1abii 2672 1  |-  ( ( U. A  i^i  B
)  =/=  (/)  <->  E. x  e.  A  ( x  i^i  B )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188    = wceq 1444    =/= wne 2622   A.wral 2737   E.wrex 2738   _Vcvv 3045    \ cdif 3401    i^i cin 3403    C_ wss 3404   (/)c0 3731   U.cuni 4198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-v 3047  df-dif 3407  df-in 3411  df-ss 3418  df-nul 3732  df-uni 4199
This theorem is referenced by:  locfinreflem  28667
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