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Theorem uniinn0 28002
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 2631 . . . 4  |-  ( -.  ( x  i^i  B
)  =/=  (/)  <->  ( x  i^i  B )  =  (/) )
21ralbii 2863 . . 3  |-  ( A. x  e.  A  -.  ( x  i^i  B )  =/=  (/)  <->  A. x  e.  A  ( x  i^i  B )  =  (/) )
3 ralnex 2878 . . 3  |-  ( A. x  e.  A  -.  ( x  i^i  B )  =/=  (/)  <->  -.  E. x  e.  A  ( x  i^i  B )  =/=  (/) )
4 unissb 4253 . . . 4  |-  ( U. A  C_  ( _V  \  B )  <->  A. x  e.  A  x  C_  ( _V  \  B ) )
5 disj2 3846 . . . 4  |-  ( ( U. A  i^i  B
)  =  (/)  <->  U. A  C_  ( _V  \  B ) )
6 disj2 3846 . . . . 5  |-  ( ( x  i^i  B )  =  (/)  <->  x  C_  ( _V 
\  B ) )
76ralbii 2863 . . . 4  |-  ( A. x  e.  A  (
x  i^i  B )  =  (/)  <->  A. x  e.  A  x  C_  ( _V  \  B ) )
84, 5, 73bitr4ri 281 . . 3  |-  ( A. x  e.  A  (
x  i^i  B )  =  (/)  <->  ( U. A  i^i  B )  =  (/) )
92, 3, 83bitr3i 278 . 2  |-  ( -. 
E. x  e.  A  ( x  i^i  B )  =/=  (/)  <->  ( U. A  i^i  B )  =  (/) )
109necon1abii 2693 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 187    = wceq 1437    =/= wne 2625   A.wral 2782   E.wrex 2783   _Vcvv 3087    \ cdif 3439    i^i cin 3441    C_ wss 3442   (/)c0 3767   U.cuni 4222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-v 3089  df-dif 3445  df-in 3449  df-ss 3456  df-nul 3768  df-uni 4223
This theorem is referenced by:  locfinreflem  28506
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