MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  disjxsn Unicode version

Theorem disjxsn 4033
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 4011 . 2  |-  (Disj  x  e.  { A } B  <->  A. y E* x ( x  e.  { A }  /\  y  e.  B
) )
2 moeq 2954 . . 3  |-  E* x  x  =  A
3 elsni 3677 . . . . 5  |-  ( x  e.  { A }  ->  x  =  A )
43adantr 451 . . . 4  |-  ( ( x  e.  { A }  /\  y  e.  B
)  ->  x  =  A )
54moimi 2203 . . 3  |-  ( E* x  x  =  A  ->  E* x ( x  e.  { A }  /\  y  e.  B
) )
62, 5ax-mp 8 . 2  |-  E* x
( x  e.  { A }  /\  y  e.  B )
71, 6mpgbir 1540 1  |- Disj  x  e. 
{ A } B
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696   E*wmo 2157   {csn 3653  Disj wdisj 4009
This theorem is referenced by:  disjx0  4034  disjdifprg  23367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rmo 2564  df-v 2803  df-sn 3659  df-disj 4010
  Copyright terms: Public domain W3C validator