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Theorem dm0 5010
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 3720 . 2  |-  ( dom  (/)  =  (/)  <->  A. x  -.  x  e.  dom  (/) )
2 noel 3708 . . . 4  |-  -.  <. x ,  y >.  e.  (/)
32nex 1672 . . 3  |-  -.  E. y <. x ,  y
>.  e.  (/)
4 vex 3025 . . . 4  |-  x  e. 
_V
54eldm2 4995 . . 3  |-  ( x  e.  dom  (/)  <->  E. y <. x ,  y >.  e.  (/) )
63, 5mtbir 300 . 2  |-  -.  x  e.  dom  (/)
71, 6mpgbir 1667 1  |-  dom  (/)  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1437   E.wex 1657    e. wcel 1872   (/)c0 3704   <.cop 3947   dom cdm 4796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-dm 4806
This theorem is referenced by:  dmxpid  5016  rn0  5048  dmxpss  5230  fn0  5656  f0dom0  5727  f10d  5806  f1o00  5807  0fv  5858  1stval  6753  bropopvvv  6831  supp0  6874  tz7.44lem1  7078  tz7.44-2  7080  tz7.44-3  7081  oicl  7997  oif  7998  swrd0  12736  dmtrclfv  13026  strlemor0  15159  symgsssg  17051  symgfisg  17052  psgnunilem5  17078  dvbsss  22799  perfdvf  22800  uhgra0  24978  umgra0  24994  clwwlknprop  25442  2wlkonot3v  25545  2spthonot3v  25546  eupa0  25644  ismgmOLD  25990  dmadjrnb  27501  f1ocnt  28326  mbfmcst  29033  0rrv  29236  conrel2d  36169  iblempty  37725  uhgr0e  38939  uhgr0  38941  usgr0  39072  0grsubgr  39097  uhg0e  39289
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