MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfdisj2 Structured version   Unicode version

Theorem dfdisj2 4409
Description: Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
Assertion
Ref Expression
dfdisj2  |-  (Disj  x  e.  A  B  <->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
Distinct variable groups:    x, y    y, A    y, B
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem dfdisj2
StepHypRef Expression
1 df-disj 4408 . 2  |-  (Disj  x  e.  A  B  <->  A. y E* x  e.  A  y  e.  B )
2 df-rmo 2801 . . 3  |-  ( E* x  e.  A  y  e.  B  <->  E* x
( x  e.  A  /\  y  e.  B
) )
32albii 1627 . 2  |-  ( A. y E* x  e.  A  y  e.  B  <->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
41, 3bitri 249 1  |-  (Disj  x  e.  A  B  <->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   A.wal 1381    e. wcel 1804   E*wmo 2269   E*wrmo 2796  Disj wdisj 4407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618
This theorem depends on definitions:  df-bi 185  df-rmo 2801  df-disj 4408
This theorem is referenced by:  disjss1  4413  nfdisj  4419  disjmoOLD  4422  invdisj  4426  disjiunOLD  4428  sndisj  4429  disjxsn  4431  disjss3  4436  vitalilem3  21997
  Copyright terms: Public domain W3C validator