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Theorem neiopne 25051
Description: If an intersection is not empty its operands are not empty. (Contributed by FL, 27-Apr-2008.)
Assertion
Ref Expression
neiopne  |-  ( ( A  i^i  B )  =/=  (/)  ->  ( A  =/=  (/)  /\  B  =/=  (/) ) )

Proof of Theorem neiopne
StepHypRef Expression
1 ineq1 3363 . . . . 5  |-  ( A  =  (/)  ->  ( A  i^i  B )  =  ( (/)  i^i  B ) )
2 incom 3361 . . . . 5  |-  ( (/)  i^i 
B )  =  ( B  i^i  (/) )
3 eqtr 2300 . . . . . 6  |-  ( ( ( A  i^i  B
)  =  ( (/)  i^i 
B )  /\  ( (/) 
i^i  B )  =  ( B  i^i  (/) ) )  ->  ( A  i^i  B )  =  ( B  i^i  (/) ) )
4 in0 3480 . . . . . 6  |-  ( B  i^i  (/) )  =  (/)
53, 4syl6eq 2331 . . . . 5  |-  ( ( ( A  i^i  B
)  =  ( (/)  i^i 
B )  /\  ( (/) 
i^i  B )  =  ( B  i^i  (/) ) )  ->  ( A  i^i  B )  =  (/) )
61, 2, 5sylancl 643 . . . 4  |-  ( A  =  (/)  ->  ( A  i^i  B )  =  (/) )
7 ineq2 3364 . . . . 5  |-  ( B  =  (/)  ->  ( A  i^i  B )  =  ( A  i^i  (/) ) )
8 in0 3480 . . . . 5  |-  ( A  i^i  (/) )  =  (/)
97, 8syl6eq 2331 . . . 4  |-  ( B  =  (/)  ->  ( A  i^i  B )  =  (/) )
106, 9jaoi 368 . . 3  |-  ( ( A  =  (/)  \/  B  =  (/) )  ->  ( A  i^i  B )  =  (/) )
1110necon3ai 2486 . 2  |-  ( ( A  i^i  B )  =/=  (/)  ->  -.  ( A  =  (/)  \/  B  =  (/) ) )
12 neanior 2531 . 2  |-  ( ( A  =/=  (/)  /\  B  =/=  (/) )  <->  -.  ( A  =  (/)  \/  B  =  (/) ) )
1311, 12sylibr 203 1  |-  ( ( A  i^i  B )  =/=  (/)  ->  ( A  =/=  (/)  /\  B  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    =/= wne 2446    i^i cin 3151   (/)c0 3455
This theorem is referenced by:  hdrmp  25706
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-in 3159  df-nul 3456
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