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Theorem dm0 5083
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 3642 . 2  |-  ( dom  (/)  =  (/)  <->  A. x  -.  x  e.  dom  (/) )
2 noel 3632 . . . 4  |-  -.  <. x ,  y >.  e.  (/)
32nex 1564 . . 3  |-  -.  E. y <. x ,  y
>.  e.  (/)
4 vex 2959 . . . 4  |-  x  e. 
_V
54eldm2 5068 . . 3  |-  ( x  e.  dom  (/)  <->  E. y <. x ,  y >.  e.  (/) )
63, 5mtbir 291 . 2  |-  -.  x  e.  dom  (/)
71, 6mpgbir 1559 1  |-  dom  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3   E.wex 1550    = wceq 1652    e. wcel 1725   (/)c0 3628   <.cop 3817   dom cdm 4878
This theorem is referenced by:  dmxpid  5089  rn0  5127  dmxpss  5300  fn0  5564  f1o00  5710  fv01  5763  1stval  6351  bropopvvv  6426  tz7.44lem1  6663  tz7.44-2  6665  tz7.44-3  6666  oicl  7498  oif  7499  strlemor0  13555  dvbsss  19789  perfdvf  19790  uhgra0  21344  umgra0  21360  usgra0  21390  eupa0  21696  ismgm  21908  dmadjrnb  23409  mbfmcst  24609  0rrv  24709  symgsssg  27385  symgfisg  27386  psgnunilem5  27394  swrd0  28183  2wlkonot3v  28342  2spthonot3v  28343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-dm 4888
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