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Theorem disjne 3858
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 3853 . . 3  |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  A  -.  x  e.  B
)
2 eleq1 2515 . . . . . 6  |-  ( x  =  C  ->  (
x  e.  B  <->  C  e.  B ) )
32notbid 294 . . . . 5  |-  ( x  =  C  ->  ( -.  x  e.  B  <->  -.  C  e.  B ) )
43rspccva 3195 . . . 4  |-  ( ( A. x  e.  A  -.  x  e.  B  /\  C  e.  A
)  ->  -.  C  e.  B )
5 eleq1a 2526 . . . . 5  |-  ( D  e.  B  ->  ( C  =  D  ->  C  e.  B ) )
65necon3bd 2655 . . . 4  |-  ( D  e.  B  ->  ( -.  C  e.  B  ->  C  =/=  D ) )
74, 6syl5com 30 . . 3  |-  ( ( A. x  e.  A  -.  x  e.  B  /\  C  e.  A
)  ->  ( D  e.  B  ->  C  =/= 
D ) )
81, 7sylanb 472 . 2  |-  ( ( ( A  i^i  B
)  =  (/)  /\  C  e.  A )  ->  ( D  e.  B  ->  C  =/=  D ) )
983impia 1194 1  |-  ( ( ( A  i^i  B
)  =  (/)  /\  C  e.  A  /\  D  e.  B )  ->  C  =/=  D )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793    i^i cin 3460   (/)c0 3770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-v 3097  df-dif 3464  df-in 3468  df-nul 3771
This theorem is referenced by:  brdom7disj  8912  brdom6disj  8913  frlmssuvc1  18802  frlmssuvc1OLD  18804  frlmsslsp  18806  frlmsslspOLD  18807  kelac1  30984
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