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

Definition df-disj 4175
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  set  x
2 cA . . 3  class  A
3 cB . . 3  class  B
41, 2, 3wdisj 4174 . 2  wff Disj  x  e.  A B
5 vy . . . . . 6  set  y
65cv 1651 . . . . 5  class  y
76, 3wcel 1725 . . . 4  wff  y  e.  B
87, 1, 2wrmo 2700 . . 3  wff  E* x  e.  A y  e.  B
98, 5wal 1549 . 2  wff  A. y E* x  e.  A
y  e.  B
104, 9wb 177 1  wff  (Disj  x  e.  A B  <->  A. y E* x  e.  A
y  e.  B )
Colors of variables: wff set class
This definition is referenced by:  dfdisj2  4176  disjss2  4177  cbvdisj  4184  nfdisj1  4187  disjor  4188  disjiun  4194  cbvdisjf  24003  disjss1f  24004  disjorf  24009  disjin  24015  disjrdx  24019
  Copyright terms: Public domain W3C validator