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Theorem disjxsn 3985
Description: A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjxsn  |- Disj  x  e. 
{ A } B
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem disjxsn
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfdisj2 3966 . 2  |-  (Disj  x  e.  { A } B  <->  A. y E* x ( x  e.  { A }  /\  y  e.  B
) )
2 moeq 2905 . . 3  |-  E* x  x  =  A
3 elsni 3599 . . . . 5  |-  ( x  e.  { A }  ->  x  =  A )
43adantr 274 . . . 4  |-  ( ( x  e.  { A }  /\  y  e.  B
)  ->  x  =  A )
54moimi 2084 . . 3  |-  ( E* x  x  =  A  ->  E* x ( x  e.  { A }  /\  y  e.  B
) )
62, 5ax-mp 5 . 2  |-  E* x
( x  e.  { A }  /\  y  e.  B )
71, 6mpgbir 1446 1  |- Disj  x  e. 
{ A } B
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1348   E*wmo 2020    e. wcel 2141   {csn 3581  Disj wdisj 3964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rmo 2456  df-v 2732  df-sn 3587  df-disj 3965
This theorem is referenced by:  disjx0  3986
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