ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  disjne Unicode version

Theorem disjne 3386
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 3381 . . 3  |-  ( ( A  i^i  B )  =  (/)  <->  A. x  e.  A  -.  x  e.  B
)
2 eleq1 2180 . . . . . 6  |-  ( x  =  C  ->  (
x  e.  B  <->  C  e.  B ) )
32notbid 641 . . . . 5  |-  ( x  =  C  ->  ( -.  x  e.  B  <->  -.  C  e.  B ) )
43rspccva 2762 . . . 4  |-  ( ( A. x  e.  A  -.  x  e.  B  /\  C  e.  A
)  ->  -.  C  e.  B )
5 eleq1a 2189 . . . . 5  |-  ( D  e.  B  ->  ( C  =  D  ->  C  e.  B ) )
65necon3bd 2328 . . . 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 282 . 2  |-  ( ( ( A  i^i  B
)  =  (/)  /\  C  e.  A )  ->  ( D  e.  B  ->  C  =/=  D ) )
983impia 1163 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 103    /\ w3a 947    = wceq 1316    e. wcel 1465    =/= wne 2285   A.wral 2393    i^i cin 3040   (/)c0 3333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-v 2662  df-dif 3043  df-in 3047  df-nul 3334
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator