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Theorem sndisj 4015
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 3995 . 2  |-  (Disj  x  e.  A { x }  <->  A. y E* x ( x  e.  A  /\  y  e.  { x } ) )
2 moeq 2941 . . 3  |-  E* x  x  =  y
3 simpr 447 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  { x } )  ->  y  e.  { x } )
4 elsn 3655 . . . . . 6  |-  ( y  e.  { x }  <->  y  =  x )
53, 4sylib 188 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  { x } )  ->  y  =  x )
65eqcomd 2288 . . . 4  |-  ( ( x  e.  A  /\  y  e.  { x } )  ->  x  =  y )
76moimi 2190 . . 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 1537 1  |- Disj  x  e.  A { x }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   E*wmo 2144   {csn 3640  Disj wdisj 3993
This theorem is referenced by:  0disj  4016
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-rmo 2551  df-v 2790  df-sn 3646  df-disj 3994
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