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

Theorem dfdisj2 4388
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 4387 . 2  |-  (Disj  x  e.  A  B  <->  A. y E* x  e.  A  y  e.  B )
2 df-rmo 2756 . . 3  |-  ( E* x  e.  A  y  e.  B  <->  E* x
( x  e.  A  /\  y  e.  B
) )
32albii 1701 . 2  |-  ( A. y E* x  e.  A  y  e.  B  <->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
41, 3bitri 257 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 189    /\ wa 375   A.wal 1452    e. wcel 1897   E*wmo 2310   E*wrmo 2751  Disj wdisj 4386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692
This theorem depends on definitions:  df-bi 190  df-rmo 2756  df-disj 4387
This theorem is referenced by:  disjss1  4392  nfdisj  4398  invdisj  4404  sndisj  4407  disjxsn  4409  disjss3  4414  vitalilem3  22616
  Copyright terms: Public domain W3C validator