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Theorem sndisj 4196
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 4176 . 2  |-  (Disj  x  e.  A { x }  <->  A. y E* x ( x  e.  A  /\  y  e.  { x } ) )
2 moeq 3102 . . 3  |-  E* x  x  =  y
3 simpr 448 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  { x } )  ->  y  e.  { x } )
4 elsn 3821 . . . . . 6  |-  ( y  e.  { x }  <->  y  =  x )
53, 4sylib 189 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  { x } )  ->  y  =  x )
65eqcomd 2440 . . . 4  |-  ( ( x  e.  A  /\  y  e.  { x } )  ->  x  =  y )
76moimi 2327 . . 3  |-  ( E* x  x  =  y  ->  E* x ( x  e.  A  /\  y  e.  { x } ) )
82, 7ax-mp 8 . 2  |-  E* x
( x  e.  A  /\  y  e.  { x } )
91, 8mpgbir 1559 1  |- Disj  x  e.  A { x }
Colors of variables: wff set class
Syntax hints:    /\ wa 359    e. wcel 1725   E*wmo 2281   {csn 3806  Disj wdisj 4174
This theorem is referenced by:  0disj  4197  sibfof  24646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-rmo 2705  df-v 2950  df-sn 3812  df-disj 4175
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