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Theorem disjne 3500
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 3495 . . 3  |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  A  -.  x  e.  B
)
2 eleq1 2343 . . . . . 6  |-  ( x  =  C  ->  (
x  e.  B  <->  C  e.  B ) )
32notbid 285 . . . . 5  |-  ( x  =  C  ->  ( -.  x  e.  B  <->  -.  C  e.  B ) )
43rspccva 2883 . . . 4  |-  ( ( A. x  e.  A  -.  x  e.  B  /\  C  e.  A
)  ->  -.  C  e.  B )
5 eleq1a 2352 . . . . 5  |-  ( D  e.  B  ->  ( C  =  D  ->  C  e.  B ) )
65necon3bd 2483 . . . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    i^i cin 3151   (/)c0 3455
This theorem is referenced by:  brdom7disj  8156  brdom6disj  8157  kelac1  27161  frlmssuvc1  27246  frlmsslsp  27248
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-v 2790  df-dif 3155  df-in 3159  df-nul 3456
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