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Theorem dm0 5023
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 3585 . 2  |-  ( dom  (/)  =  (/)  <->  A. x  -.  x  e.  dom  (/) )
2 noel 3575 . . . 4  |-  -.  <. x ,  y >.  e.  (/)
32nex 1561 . . 3  |-  -.  E. y <. x ,  y
>.  e.  (/)
4 vex 2902 . . . 4  |-  x  e. 
_V
54eldm2 5008 . . 3  |-  ( x  e.  dom  (/)  <->  E. y <. x ,  y >.  e.  (/) )
63, 5mtbir 291 . 2  |-  -.  x  e.  dom  (/)
71, 6mpgbir 1556 1  |-  dom  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3   E.wex 1547    = wceq 1649    e. wcel 1717   (/)c0 3571   <.cop 3760   dom cdm 4818
This theorem is referenced by:  dmxpid  5029  rn0  5067  dmxpss  5240  fn0  5504  f1o00  5650  fv01  5702  1stval  6290  bropopvvv  6365  tz7.44lem1  6599  tz7.44-2  6601  tz7.44-3  6602  oicl  7431  oif  7432  strlemor0  13482  dvbsss  19656  perfdvf  19657  uhgra0  21211  umgra0  21227  usgra0  21257  eupa0  21544  ismgm  21756  dmadjrnb  23257  mbfmcst  24403  0rrv  24488  symgsssg  27077  symgfisg  27078  psgnunilem5  27086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-dm 4828
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