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Theorem disjel 3666
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 3664 . . 3  |-  ( ( A  i^i  B )  =  (/)  <->  A  =  ( A  \  B ) )
2 eleq2 2496 . . . 4  |-  ( A  =  ( A  \  B )  ->  ( C  e.  A  <->  C  e.  ( A  \  B ) ) )
3 eldifn 3462 . . . 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 1652    e. wcel 1725    \ cdif 3309    i^i cin 3311   (/)c0 3620
This theorem is referenced by:  disjxun  4202  fvun1  5786  dedekindle  25180  fprodsplit  25281
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-v 2950  df-dif 3315  df-in 3319  df-nul 3621
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