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Theorem dm0 5221
 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

Proof of Theorem dm0
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eq0 3800 . 2
2 noel 3788 . . . 4
32nex 1627 . . 3
4 vex 3112 . . . 4
54eldm2 5206 . . 3
63, 5mtbir 299 . 2
71, 6mpgbir 1622 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  =wceq 1395  E.wex 1612  e.wcel 1818   c0 3784  <.cop 4035  domcdm 5004 This theorem is referenced by:  dmxpid  5227  rn0  5259  dmxpss  5443  fn0  5705  f0dom0  5774  f1o00  5853  0fv  5904  1stval  6802  bropopvvv  6880  supp0  6923  tz7.44lem1  7090  tz7.44-2  7092  tz7.44-3  7093  oicl  7975  oif  7976  swrd0  12658  strlemor0  14723  symgsssg  16492  symgfisg  16493  psgnunilem5  16519  dvbsss  22306  perfdvf  22307  uhgra0  24309  umgra0  24325  usgra0  24370  clwwlknprop  24772  2wlkonot3v  24875  2spthonot3v  24876  eupa0  24974  ismgmOLD  25322  dmadjrnb  26825  mbfmcst  28230  0rrv  28390  iblempty  31764  uhg0e  32376  conrel2d  37762 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  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 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-dm 5014
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