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

Definition df-disj 4418
Description: A collection of classes  B (
x ) is disjoint when for each element  y, it is in  B ( x ) for at most one  x. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.)
Assertion
Ref Expression
df-disj  |-  (Disj  x  e.  A  B  <->  A. y E* x  e.  A  y  e.  B )
Distinct variable groups:    x, y    y, A    y, B
Allowed substitution hints:    A( x)    B( x)

Detailed syntax breakdown of Definition df-disj
StepHypRef Expression
1 vx . . 3  setvar  x
2 cA . . 3  class  A
3 cB . . 3  class  B
41, 2, 3wdisj 4417 . 2  wff Disj  x  e.  A  B
5 vy . . . . . 6  setvar  y
65cv 1378 . . . . 5  class  y
76, 3wcel 1767 . . . 4  wff  y  e.  B
87, 1, 2wrmo 2817 . . 3  wff  E* x  e.  A  y  e.  B
98, 5wal 1377 . 2  wff  A. y E* x  e.  A  y  e.  B
104, 9wb 184 1  wff  (Disj  x  e.  A  B  <->  A. y E* x  e.  A  y  e.  B )
Colors of variables: wff setvar class
This definition is referenced by:  dfdisj2  4419  disjss2  4420  cbvdisj  4427  nfdisj1  4430  disjor  4431  disjiun  4437  cbvdisjf  27107  disjss1f  27108  disjorf  27113  disjin  27119  disjrdx  27123  ddemeas  27848
  Copyright terms: Public domain W3C validator