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Theorem disjxsn 4305
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 4283 . 2  |-  (Disj  x  e.  { A } B  <->  A. y E* x ( x  e.  { A }  /\  y  e.  B
) )
2 moeq 3154 . . 3  |-  E* x  x  =  A
3 elsni 3921 . . . . 5  |-  ( x  e.  { A }  ->  x  =  A )
43adantr 465 . . . 4  |-  ( ( x  e.  { A }  /\  y  e.  B
)  ->  x  =  A )
54moimi 2320 . . 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 1595 1  |- Disj  x  e. 
{ A } B
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369    e. wcel 1756   E*wmo 2254   {csn 3896  Disj wdisj 4281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-rmo 2742  df-v 2993  df-sn 3897  df-disj 4282
This theorem is referenced by:  disjx0  4306  disjdifprg  25938
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