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Theorem dm0 4577
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 3273 . 2  |-  ( dom  (/)  =  (/)  <->  A. x  -.  x  e.  dom  (/) )
2 noel 3262 . . . 4  |-  -.  <. x ,  y >.  e.  (/)
32nex 1430 . . 3  |-  -.  E. y <. x ,  y
>.  e.  (/)
4 vex 2605 . . . 4  |-  x  e. 
_V
54eldm2 4561 . . 3  |-  ( x  e.  dom  (/)  <->  E. y <. x ,  y >.  e.  (/) )
63, 5mtbir 629 . 2  |-  -.  x  e.  dom  (/)
71, 6mpgbir 1383 1  |-  dom  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1285   E.wex 1422    e. wcel 1434   (/)c0 3258   <.cop 3409   dom cdm 4371
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-dif 2976  df-un 2978  df-nul 3259  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-dm 4381
This theorem is referenced by:  rn0  4616  fn0  5049  f1o00  5192  rdg0  6036  frec0g  6046
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