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Theorem 0pssin 36365
Description: Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
Assertion
Ref Expression
0pssin  |-  ( (/)  C.  ( A  i^i  B
)  <->  E. x ( x  e.  A  /\  x  e.  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem 0pssin
StepHypRef Expression
1 0pss 3802 . 2  |-  ( (/)  C.  ( A  i^i  B
)  <->  ( A  i^i  B )  =/=  (/) )
2 ndisj 36364 . 2  |-  ( ( A  i^i  B )  =/=  (/)  <->  E. x ( x  e.  A  /\  x  e.  B ) )
31, 2bitri 253 1  |-  ( (/)  C.  ( A  i^i  B
)  <->  E. x ( x  e.  A  /\  x  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371   E.wex 1663    e. wcel 1887    =/= wne 2622    i^i cin 3403    C. wpss 3405   (/)c0 3731
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-or 372  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-v 3047  df-dif 3407  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732
This theorem is referenced by: (None)
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