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

Theorem disjxsn 4206
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 4184 . 2  |-  (Disj  x  e.  { A } B  <->  A. y E* x ( x  e.  { A }  /\  y  e.  B
) )
2 moeq 3110 . . 3  |-  E* x  x  =  A
3 elsni 3838 . . . . 5  |-  ( x  e.  { A }  ->  x  =  A )
43adantr 452 . . . 4  |-  ( ( x  e.  { A }  /\  y  e.  B
)  ->  x  =  A )
54moimi 2328 . . 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 1559 1  |- Disj  x  e. 
{ A } B
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   E*wmo 2282   {csn 3814  Disj wdisj 4182
This theorem is referenced by:  disjx0  4207  disjdifprg  24017
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rmo 2713  df-v 2958  df-sn 3820  df-disj 4183
  Copyright terms: Public domain W3C validator