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Theorem dm0 5226
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 3809 . 2  |-  ( dom  (/)  =  (/)  <->  A. x  -.  x  e.  dom  (/) )
2 noel 3797 . . . 4  |-  -.  <. x ,  y >.  e.  (/)
32nex 1628 . . 3  |-  -.  E. y <. x ,  y
>.  e.  (/)
4 vex 3112 . . . 4  |-  x  e. 
_V
54eldm2 5211 . . 3  |-  ( x  e.  dom  (/)  <->  E. y <. x ,  y >.  e.  (/) )
63, 5mtbir 299 . 2  |-  -.  x  e.  dom  (/)
71, 6mpgbir 1623 1  |-  dom  (/)  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1395   E.wex 1613    e. wcel 1819   (/)c0 3793   <.cop 4038   dom cdm 5008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-dm 5018
This theorem is referenced by:  dmxpid  5232  rn0  5264  dmxpss  5445  fn0  5706  f0dom0  5775  f1o00  5854  0fv  5905  1stval  6801  bropopvvv  6879  supp0  6922  tz7.44lem1  7089  tz7.44-2  7091  tz7.44-3  7092  oicl  7972  oif  7973  swrd0  12670  strlemor0  14738  symgsssg  16619  symgfisg  16620  psgnunilem5  16646  dvbsss  22432  perfdvf  22433  uhgra0  24436  umgra0  24452  usgra0  24497  clwwlknprop  24899  2wlkonot3v  25002  2spthonot3v  25003  eupa0  25101  ismgmOLD  25449  dmadjrnb  26952  mbfmcst  28403  0rrv  28587  iblempty  31967  uhg0e  32638  conrel2d  37922  dmtrclfv  37978
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