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Theorem disjel 3417
Description: A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)
Assertion
Ref Expression
disjel  |-  ( ( ( A  i^i  B
)  =  (/)  /\  C  e.  A )  ->  -.  C  e.  B )

Proof of Theorem disjel
StepHypRef Expression
1 disj3 3415 . . 3  |-  ( ( A  i^i  B )  =  (/)  <->  A  =  ( A  \  B ) )
2 eleq2 2203 . . . 4  |-  ( A  =  ( A  \  B )  ->  ( C  e.  A  <->  C  e.  ( A  \  B ) ) )
3 eldifn 3199 . . . 4  |-  ( C  e.  ( A  \  B )  ->  -.  C  e.  B )
42, 3syl6bi 162 . . 3  |-  ( A  =  ( A  \  B )  ->  ( C  e.  A  ->  -.  C  e.  B ) )
51, 4sylbi 120 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( C  e.  A  ->  -.  C  e.  B )
)
65imp 123 1  |-  ( ( ( A  i^i  B
)  =  (/)  /\  C  e.  A )  ->  -.  C  e.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480    \ cdif 3068    i^i cin 3070   (/)c0 3363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-dif 3073  df-in 3077  df-nul 3364
This theorem is referenced by:  fvun1  5490  ctssdccl  6999  fsumsplit  11200
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