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Theorem dfnul3 3534
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 pm3.24 852 . . . . 5  |-  -.  (
x  e.  A  /\  -.  x  e.  A
)
2 eqid 2358 . . . . 5  |-  x  =  x
31, 22th 230 . . . 4  |-  ( -.  ( x  e.  A  /\  -.  x  e.  A
)  <->  x  =  x
)
43con1bii 321 . . 3  |-  ( -.  x  =  x  <->  ( x  e.  A  /\  -.  x  e.  A ) )
54abbii 2470 . 2  |-  { x  |  -.  x  =  x }  =  { x  |  ( x  e.  A  /\  -.  x  e.  A ) }
6 dfnul2 3533 . 2  |-  (/)  =  {
x  |  -.  x  =  x }
7 df-rab 2628 . 2  |-  { x  e.  A  |  -.  x  e.  A }  =  { x  |  ( x  e.  A  /\  -.  x  e.  A
) }
85, 6, 73eqtr4i 2388 1  |-  (/)  =  {
x  e.  A  |  -.  x  e.  A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1642    e. wcel 1710   {cab 2344   {crab 2623   (/)c0 3531
This theorem is referenced by:  difidALT  3599  kmlem3  7868
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rab 2628  df-v 2866  df-dif 3231  df-nul 3532
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