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Theorem dm0 4638
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 3299 . 2  |-  ( dom  (/)  =  (/)  <->  A. x  -.  x  e.  dom  (/) )
2 noel 3288 . . . 4  |-  -.  <. x ,  y >.  e.  (/)
32nex 1434 . . 3  |-  -.  E. y <. x ,  y
>.  e.  (/)
4 vex 2622 . . . 4  |-  x  e. 
_V
54eldm2 4622 . . 3  |-  ( x  e.  dom  (/)  <->  E. y <. x ,  y >.  e.  (/) )
63, 5mtbir 631 . 2  |-  -.  x  e.  dom  (/)
71, 6mpgbir 1387 1  |-  dom  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1289   E.wex 1426    e. wcel 1438   (/)c0 3284   <.cop 3444   dom cdm 4428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-un 3001  df-nul 3285  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-dm 4438
This theorem is referenced by:  rn0  4677  fn0  5119  f0dom0  5188  f1o00  5272  rdg0  6134  frec0g  6144
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