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Theorem disjne 3324
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 3319 . . 3  |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  A  -.  x  e.  B
)
2 eleq1 2147 . . . . . 6  |-  ( x  =  C  ->  (
x  e.  B  <->  C  e.  B ) )
32notbid 625 . . . . 5  |-  ( x  =  C  ->  ( -.  x  e.  B  <->  -.  C  e.  B ) )
43rspccva 2714 . . . 4  |-  ( ( A. x  e.  A  -.  x  e.  B  /\  C  e.  A
)  ->  -.  C  e.  B )
5 eleq1a 2156 . . . . 5  |-  ( D  e.  B  ->  ( C  =  D  ->  C  e.  B ) )
65necon3bd 2294 . . . 4  |-  ( D  e.  B  ->  ( -.  C  e.  B  ->  C  =/=  D ) )
74, 6syl5com 29 . . 3  |-  ( ( A. x  e.  A  -.  x  e.  B  /\  C  e.  A
)  ->  ( D  e.  B  ->  C  =/= 
D ) )
81, 7sylanb 278 . 2  |-  ( ( ( A  i^i  B
)  =  (/)  /\  C  e.  A )  ->  ( D  e.  B  ->  C  =/=  D ) )
983impia 1138 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 102    /\ w3a 922    = wceq 1287    e. wcel 1436    =/= wne 2251   A.wral 2355    i^i cin 2987   (/)c0 3275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-v 2617  df-dif 2990  df-in 2994  df-nul 3276
This theorem is referenced by: (None)
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