MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfnul3 Unicode version

Theorem dfnul3 3599
Description: Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
Assertion
Ref Expression
dfnul3  |-  (/)  =  {
x  e.  A  |  -.  x  e.  A }

Proof of Theorem dfnul3
StepHypRef Expression
1 pm3.24 853 . . . . 5  |-  -.  (
x  e.  A  /\  -.  x  e.  A
)
2 equid 1684 . . . . 5  |-  x  =  x
31, 22th 231 . . . 4  |-  ( -.  ( x  e.  A  /\  -.  x  e.  A
)  <->  x  =  x
)
43con1bii 322 . . 3  |-  ( -.  x  =  x  <->  ( x  e.  A  /\  -.  x  e.  A ) )
54abbii 2524 . 2  |-  { x  |  -.  x  =  x }  =  { x  |  ( x  e.  A  /\  -.  x  e.  A ) }
6 dfnul2 3598 . 2  |-  (/)  =  {
x  |  -.  x  =  x }
7 df-rab 2683 . 2  |-  { x  e.  A  |  -.  x  e.  A }  =  { x  |  ( x  e.  A  /\  -.  x  e.  A
) }
85, 6, 73eqtr4i 2442 1  |-  (/)  =  {
x  e.  A  |  -.  x  e.  A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2398   {crab 2678   (/)c0 3596
This theorem is referenced by:  difidALT  3665  kmlem3  7996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rab 2683  df-v 2926  df-dif 3291  df-nul 3597
  Copyright terms: Public domain W3C validator