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Theorem disjsn 3524
Description: Intersection with the singleton of a non-member is disjoint. (Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
Assertion
Ref Expression
disjsn  |-  ( ( A  i^i  { B } )  =  (/)  <->  -.  B  e.  A )

Proof of Theorem disjsn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 disj1 3352 . 2  |-  ( ( A  i^i  { B } )  =  (/)  <->  A. x ( x  e.  A  ->  -.  x  e.  { B } ) )
2 con2b 631 . . . 4  |-  ( ( x  e.  A  ->  -.  x  e.  { B } )  <->  ( x  e.  { B }  ->  -.  x  e.  A ) )
3 velsn 3483 . . . . 5  |-  ( x  e.  { B }  <->  x  =  B )
43imbi1i 237 . . . 4  |-  ( ( x  e.  { B }  ->  -.  x  e.  A )  <->  ( x  =  B  ->  -.  x  e.  A ) )
5 imnan 662 . . . 4  |-  ( ( x  =  B  ->  -.  x  e.  A
)  <->  -.  ( x  =  B  /\  x  e.  A ) )
62, 4, 53bitri 205 . . 3  |-  ( ( x  e.  A  ->  -.  x  e.  { B } )  <->  -.  (
x  =  B  /\  x  e.  A )
)
76albii 1411 . 2  |-  ( A. x ( x  e.  A  ->  -.  x  e.  { B } )  <->  A. x  -.  (
x  =  B  /\  x  e.  A )
)
8 alnex 1440 . . 3  |-  ( A. x  -.  ( x  =  B  /\  x  e.  A )  <->  -.  E. x
( x  =  B  /\  x  e.  A
) )
9 df-clel 2091 . . 3  |-  ( B  e.  A  <->  E. x
( x  =  B  /\  x  e.  A
) )
108, 9xchbinxr 646 . 2  |-  ( A. x  -.  ( x  =  B  /\  x  e.  A )  <->  -.  B  e.  A )
111, 7, 103bitri 205 1  |-  ( ( A  i^i  { B } )  =  (/)  <->  -.  B  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1294    = wceq 1296   E.wex 1433    e. wcel 1445    i^i cin 3012   (/)c0 3302   {csn 3466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-v 2635  df-dif 3015  df-in 3019  df-nul 3303  df-sn 3472
This theorem is referenced by:  disjsn2  3525  ssdifsn  3590  orddisj  4390  ndmima  4842  funtpg  5099  fnunsn  5155  ressnop0  5517  ftpg  5520  fsnunf  5536  fsnunfv  5537  phpm  6661  fiunsnnn  6677  ac6sfi  6694  unsnfi  6709  tpfidisj  6718  iunfidisj  6735  pm54.43  6915  fzpreddisj  9634  fzp1disj  9643  frecfzennn  9982  hashunsng  10346  hashxp  10365  fsumsplitsn  10969  sumtp  10973  fsumsplitsnun  10978  fsum2dlemstep  10993  fsumconst  11013  fsumabs  11024  fsumiun  11036  structcnvcnv  11675
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