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Theorem dm0 5216
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 3800 . 2  |-  ( dom  (/)  =  (/)  <->  A. x  -.  x  e.  dom  (/) )
2 noel 3789 . . . 4  |-  -.  <. x ,  y >.  e.  (/)
32nex 1610 . . 3  |-  -.  E. y <. x ,  y
>.  e.  (/)
4 vex 3116 . . . 4  |-  x  e. 
_V
54eldm2 5201 . . 3  |-  ( x  e.  dom  (/)  <->  E. y <. x ,  y >.  e.  (/) )
63, 5mtbir 299 . 2  |-  -.  x  e.  dom  (/)
71, 6mpgbir 1605 1  |-  dom  (/)  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379   E.wex 1596    e. wcel 1767   (/)c0 3785   <.cop 4033   dom cdm 4999
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-or 370  df-an 371  df-3an 975  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-ne 2664  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-dm 5009
This theorem is referenced by:  dmxpid  5222  rn0  5254  dmxpss  5438  fn0  5700  f0dom0  5769  f1o00  5848  0fv  5899  1stval  6787  bropopvvv  6864  supp0  6907  tz7.44lem1  7072  tz7.44-2  7074  tz7.44-3  7075  oicl  7955  oif  7956  swrd0  12624  strlemor0  14584  symgsssg  16307  symgfisg  16308  psgnunilem5  16334  dvbsss  22133  perfdvf  22134  uhgra0  24082  umgra0  24098  usgra0  24143  clwwlknprop  24545  2wlkonot3v  24648  2spthonot3v  24649  eupa0  24747  ismgm  25095  dmadjrnb  26598  mbfmcst  27981  0rrv  28141  iblempty  31510  uhg0e  32070  conrel2d  36999
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