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Theorem disjel 3617
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 3615 . . 3  |-  ( ( A  i^i  B )  =  (/)  <->  A  =  ( A  \  B ) )
2 eleq2 2448 . . . 4  |-  ( A  =  ( A  \  B )  ->  ( C  e.  A  <->  C  e.  ( A  \  B ) ) )
3 eldifn 3413 . . . 4  |-  ( C  e.  ( A  \  B )  ->  -.  C  e.  B )
42, 3syl6bi 220 . . 3  |-  ( A  =  ( A  \  B )  ->  ( C  e.  A  ->  -.  C  e.  B ) )
51, 4sylbi 188 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( C  e.  A  ->  -.  C  e.  B )
)
65imp 419 1  |-  ( ( ( A  i^i  B
)  =  (/)  /\  C  e.  A )  ->  -.  C  e.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    \ cdif 3260    i^i cin 3262   (/)c0 3571
This theorem is referenced by:  disjxun  4151  fvun1  5733  dedekindle  24967  fprodsplit  25068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-v 2901  df-dif 3266  df-in 3270  df-nul 3572
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