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

Theorem sndisj 3996
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 3979 . 2  |-  (Disj  x  e.  A  { x } 
<-> 
A. y E* x
( x  e.  A  /\  y  e.  { x } ) )
2 moeq 2912 . . 3  |-  E* x  x  =  y
3 simpr 110 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  { x } )  ->  y  e.  { x } )
4 velsn 3608 . . . . . 6  |-  ( y  e.  { x }  <->  y  =  x )
53, 4sylib 122 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  { x } )  ->  y  =  x )
65eqcomd 2183 . . . 4  |-  ( ( x  e.  A  /\  y  e.  { x } )  ->  x  =  y )
76moimi 2091 . . 3  |-  ( E* x  x  =  y  ->  E* x ( x  e.  A  /\  y  e.  { x } ) )
82, 7ax-mp 5 . 2  |-  E* x
( x  e.  A  /\  y  e.  { x } )
91, 8mpgbir 1453 1  |- Disj  x  e.  A  { x }
Colors of variables: wff set class
Syntax hints:    /\ wa 104   E*wmo 2027    e. wcel 2148   {csn 3591  Disj wdisj 3977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rmo 2463  df-v 2739  df-sn 3597  df-disj 3978
This theorem is referenced by:  0disj  3997  disjsnxp  6232
  Copyright terms: Public domain W3C validator