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Theorem dm0 5086
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 3644 . 2  |-  ( dom  (/)  =  (/)  <->  A. x  -.  x  e.  dom  (/) )
2 noel 3634 . . . 4  |-  -.  <. x ,  y >.  e.  (/)
32nex 1565 . . 3  |-  -.  E. y <. x ,  y
>.  e.  (/)
4 vex 2961 . . . 4  |-  x  e. 
_V
54eldm2 5071 . . 3  |-  ( x  e.  dom  (/)  <->  E. y <. x ,  y >.  e.  (/) )
63, 5mtbir 292 . 2  |-  -.  x  e.  dom  (/)
71, 6mpgbir 1560 1  |-  dom  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3   E.wex 1551    = wceq 1653    e. wcel 1726   (/)c0 3630   <.cop 3819   dom cdm 4881
This theorem is referenced by:  dmxpid  5092  rn0  5130  dmxpss  5303  fn0  5567  f1o00  5713  fv01  5766  1stval  6354  bropopvvv  6429  tz7.44lem1  6666  tz7.44-2  6668  tz7.44-3  6669  oicl  7501  oif  7502  strlemor0  13560  dvbsss  19794  perfdvf  19795  uhgra0  21349  umgra0  21365  usgra0  21395  eupa0  21701  ismgm  21913  dmadjrnb  23414  mbfmcst  24614  0rrv  24714  symgsssg  27399  symgfisg  27400  psgnunilem5  27408  elovmpt3rab1  28107  swrd0  28217  2wlkonot3v  28407  2spthonot3v  28408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-dm 4891
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