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Theorem disjne 3534
Description: Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
disjne  |-  ( ( ( A  i^i  B
)  =  (/)  /\  C  e.  A  /\  D  e.  B )  ->  C  =/=  D )

Proof of Theorem disjne
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 disj 3529 . . 3  |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  A  -.  x  e.  B
)
2 eleq1 2376 . . . . . 6  |-  ( x  =  C  ->  (
x  e.  B  <->  C  e.  B ) )
32notbid 285 . . . . 5  |-  ( x  =  C  ->  ( -.  x  e.  B  <->  -.  C  e.  B ) )
43rspccva 2917 . . . 4  |-  ( ( A. x  e.  A  -.  x  e.  B  /\  C  e.  A
)  ->  -.  C  e.  B )
5 eleq1a 2385 . . . . 5  |-  ( D  e.  B  ->  ( C  =  D  ->  C  e.  B ) )
65necon3bd 2516 . . . 4  |-  ( D  e.  B  ->  ( -.  C  e.  B  ->  C  =/=  D ) )
74, 6syl5com 26 . . 3  |-  ( ( A. x  e.  A  -.  x  e.  B  /\  C  e.  A
)  ->  ( D  e.  B  ->  C  =/= 
D ) )
81, 7sylanb 458 . 2  |-  ( ( ( A  i^i  B
)  =  (/)  /\  C  e.  A )  ->  ( D  e.  B  ->  C  =/=  D ) )
983impia 1148 1  |-  ( ( ( A  i^i  B
)  =  (/)  /\  C  e.  A  /\  D  e.  B )  ->  C  =/=  D )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   A.wral 2577    i^i cin 3185   (/)c0 3489
This theorem is referenced by:  brdom7disj  8201  brdom6disj  8202  kelac1  26309  frlmssuvc1  26394  frlmsslsp  26396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-v 2824  df-dif 3189  df-in 3193  df-nul 3490
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