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Theorem dfnul3 3255
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 1605 . . . . 5  |-  x  =  x
21notnoti 584 . . . 4  |-  -.  -.  x  =  x
3 pm3.24 637 . . . 4  |-  -.  (
x  e.  A  /\  -.  x  e.  A
)
42, 32false 627 . . 3  |-  ( -.  x  =  x  <->  ( x  e.  A  /\  -.  x  e.  A ) )
54abbii 2169 . 2  |-  { x  |  -.  x  =  x }  =  { x  |  ( x  e.  A  /\  -.  x  e.  A ) }
6 dfnul2 3254 . 2  |-  (/)  =  {
x  |  -.  x  =  x }
7 df-rab 2332 . 2  |-  { x  e.  A  |  -.  x  e.  A }  =  { x  |  ( x  e.  A  /\  -.  x  e.  A
) }
85, 6, 73eqtr4i 2086 1  |-  (/)  =  {
x  e.  A  |  -.  x  e.  A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 101    = wceq 1259    e. wcel 1409   {cab 2042   {crab 2327   (/)c0 3252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332  df-v 2576  df-dif 2948  df-nul 3253
This theorem is referenced by:  difidALT  3321
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