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Theorem dfnul3 3788
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 pm3.24 880 . . . . 5  |-  -.  (
x  e.  A  /\  -.  x  e.  A
)
2 equid 1740 . . . . 5  |-  x  =  x
31, 22th 239 . . . 4  |-  ( -.  ( x  e.  A  /\  -.  x  e.  A
)  <->  x  =  x
)
43con1bii 331 . . 3  |-  ( -.  x  =  x  <->  ( x  e.  A  /\  -.  x  e.  A ) )
54abbii 2601 . 2  |-  { x  |  -.  x  =  x }  =  { x  |  ( x  e.  A  /\  -.  x  e.  A ) }
6 dfnul2 3787 . 2  |-  (/)  =  {
x  |  -.  x  =  x }
7 df-rab 2823 . 2  |-  { x  e.  A  |  -.  x  e.  A }  =  { x  |  ( x  e.  A  /\  -.  x  e.  A
) }
85, 6, 73eqtr4i 2506 1  |-  (/)  =  {
x  e.  A  |  -.  x  e.  A }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   {crab 2818   (/)c0 3785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-dif 3479  df-nul 3786
This theorem is referenced by:  difidALT  3896  kmlem3  8533
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