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Theorem dm0 4859
Description: The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dm0  |-  dom  (/)  =  (/)

Proof of Theorem dm0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eq0 3456 . 2  |-  ( dom  (/)  =  (/)  <->  A. x  -.  x  e.  dom  (/) )
2 noel 3441 . . . 4  |-  -.  <. x ,  y >.  e.  (/)
32nex 1511 . . 3  |-  -.  E. y <. x ,  y
>.  e.  (/)
4 vex 2755 . . . 4  |-  x  e. 
_V
54eldm2 4843 . . 3  |-  ( x  e.  dom  (/)  <->  E. y <. x ,  y >.  e.  (/) )
63, 5mtbir 672 . 2  |-  -.  x  e.  dom  (/)
71, 6mpgbir 1464 1  |-  dom  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1364   E.wex 1503    e. wcel 2160   (/)c0 3437   <.cop 3610   dom cdm 4644
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-un 3148  df-nul 3438  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-dm 4654
This theorem is referenced by:  rn0  4901  sqxpeq0  5070  fn0  5354  f0dom0  5428  f1o00  5515  rdg0  6413  frec0g  6423  ennnfonelemj0  12455  ennnfonelem1  12461  ennnfonelemkh  12466  ennnfonelemhf1o  12467
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