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Theorem uniinn0 28241
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 2647 . . . 4  |-  ( -.  ( x  i^i  B
)  =/=  (/)  <->  ( x  i^i  B )  =  (/) )
21ralbii 2823 . . 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 4221 . . . 4  |-  ( U. A  C_  ( _V  \  B )  <->  A. x  e.  A  x  C_  ( _V  \  B ) )
5 disj2 3816 . . . 4  |-  ( ( U. A  i^i  B
)  =  (/)  <->  U. A  C_  ( _V  \  B ) )
6 disj2 3816 . . . . 5  |-  ( ( x  i^i  B )  =  (/)  <->  x  C_  ( _V 
\  B ) )
76ralbii 2823 . . . 4  |-  ( A. x  e.  A  (
x  i^i  B )  =  (/)  <->  A. x  e.  A  x  C_  ( _V  \  B ) )
84, 5, 73bitr4ri 286 . . 3  |-  ( A. x  e.  A  (
x  i^i  B )  =  (/)  <->  ( U. A  i^i  B )  =  (/) )
92, 3, 83bitr3i 283 . 2  |-  ( -. 
E. x  e.  A  ( x  i^i  B )  =/=  (/)  <->  ( U. A  i^i  B )  =  (/) )
109necon1abii 2691 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 189    = wceq 1452    =/= wne 2641   A.wral 2756   E.wrex 2757   _Vcvv 3031    \ cdif 3387    i^i cin 3389    C_ wss 3390   (/)c0 3722   U.cuni 4190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-v 3033  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-uni 4191
This theorem is referenced by:  locfinreflem  28741
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