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Theorem dfnul3 3437
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 1711 . . . . 5  |-  x  =  x
21notnoti 646 . . . 4  |-  -.  -.  x  =  x
3 pm3.24 694 . . . 4  |-  -.  (
x  e.  A  /\  -.  x  e.  A
)
42, 32false 702 . . 3  |-  ( -.  x  =  x  <->  ( x  e.  A  /\  -.  x  e.  A ) )
54abbii 2303 . 2  |-  { x  |  -.  x  =  x }  =  { x  |  ( x  e.  A  /\  -.  x  e.  A ) }
6 dfnul2 3436 . 2  |-  (/)  =  {
x  |  -.  x  =  x }
7 df-rab 2474 . 2  |-  { x  e.  A  |  -.  x  e.  A }  =  { x  |  ( x  e.  A  /\  -.  x  e.  A
) }
85, 6, 73eqtr4i 2218 1  |-  (/)  =  {
x  e.  A  |  -.  x  e.  A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 104    = wceq 1363    e. wcel 2158   {cab 2173   {crab 2469   (/)c0 3434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rab 2474  df-v 2751  df-dif 3143  df-nul 3435
This theorem is referenced by:  difidALT  3504
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