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Theorem sndisj 3789
Description: Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
sndisj  |- Disj  x  e.  A  { x }

Proof of Theorem sndisj
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfdisj2 3776 . 2  |-  (Disj  x  e.  A  { x } 
<-> 
A. y E* x
( x  e.  A  /\  y  e.  { x } ) )
2 moeq 2768 . . 3  |-  E* x  x  =  y
3 simpr 108 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  { x } )  ->  y  e.  { x } )
4 velsn 3423 . . . . . 6  |-  ( y  e.  { x }  <->  y  =  x )
53, 4sylib 120 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  { x } )  ->  y  =  x )
65eqcomd 2087 . . . 4  |-  ( ( x  e.  A  /\  y  e.  { x } )  ->  x  =  y )
76moimi 2007 . . 3  |-  ( E* x  x  =  y  ->  E* x ( x  e.  A  /\  y  e.  { x } ) )
82, 7ax-mp 7 . 2  |-  E* x
( x  e.  A  /\  y  e.  { x } )
91, 8mpgbir 1383 1  |- Disj  x  e.  A  { x }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    e. wcel 1434   E*wmo 1943   {csn 3406  Disj wdisj 3774
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rmo 2357  df-v 2604  df-sn 3412  df-disj 3775
This theorem is referenced by:  0disj  3790
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