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Theorem disjxsn 4433
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 4412 . 2  |-  (Disj  x  e.  { A } B  <->  A. y E* x ( x  e.  { A }  /\  y  e.  B
) )
2 moeq 3272 . . 3  |-  E* x  x  =  A
3 elsni 4041 . . . . 5  |-  ( x  e.  { A }  ->  x  =  A )
43adantr 463 . . . 4  |-  ( ( x  e.  { A }  /\  y  e.  B
)  ->  x  =  A )
54moimi 2338 . . 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 1627 1  |- Disj  x  e. 
{ A } B
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1398    e. wcel 1823   E*wmo 2285   {csn 4016  Disj wdisj 4410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rmo 2812  df-v 3108  df-sn 4017  df-disj 4411
This theorem is referenced by:  disjx0  4434  disjdifprg  27646
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