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Theorem dfnul3 3370
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 equid 1678 . . . . 5  |-  x  =  x
21notnoti 635 . . . 4  |-  -.  -.  x  =  x
3 pm3.24 683 . . . 4  |-  -.  (
x  e.  A  /\  -.  x  e.  A
)
42, 32false 691 . . 3  |-  ( -.  x  =  x  <->  ( x  e.  A  /\  -.  x  e.  A ) )
54abbii 2256 . 2  |-  { x  |  -.  x  =  x }  =  { x  |  ( x  e.  A  /\  -.  x  e.  A ) }
6 dfnul2 3369 . 2  |-  (/)  =  {
x  |  -.  x  =  x }
7 df-rab 2426 . 2  |-  { x  e.  A  |  -.  x  e.  A }  =  { x  |  ( x  e.  A  /\  -.  x  e.  A
) }
85, 6, 73eqtr4i 2171 1  |-  (/)  =  {
x  e.  A  |  -.  x  e.  A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    = wceq 1332    e. wcel 1481   {cab 2126   {crab 2421   (/)c0 3367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rab 2426  df-v 2691  df-dif 3077  df-nul 3368
This theorem is referenced by:  difidALT  3436
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