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Theorem dm0 5164
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 3763 . 2  |-  ( dom  (/)  =  (/)  <->  A. x  -.  x  e.  dom  (/) )
2 noel 3752 . . . 4  |-  -.  <. x ,  y >.  e.  (/)
32nex 1601 . . 3  |-  -.  E. y <. x ,  y
>.  e.  (/)
4 vex 3081 . . . 4  |-  x  e. 
_V
54eldm2 5149 . . 3  |-  ( x  e.  dom  (/)  <->  E. y <. x ,  y >.  e.  (/) )
63, 5mtbir 299 . 2  |-  -.  x  e.  dom  (/)
71, 6mpgbir 1596 1  |-  dom  (/)  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1370   E.wex 1587    e. wcel 1758   (/)c0 3748   <.cop 3994   dom cdm 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-dm 4961
This theorem is referenced by:  dmxpid  5170  rn0  5202  dmxpss  5380  fn0  5641  f0dom0  5706  f1o00  5784  0fv  5835  1stval  6692  bropopvvv  6766  supp0  6808  tz7.44lem1  6974  tz7.44-2  6976  tz7.44-3  6977  oicl  7857  oif  7858  swrd0  12448  strlemor0  14386  symgsssg  16095  symgfisg  16096  psgnunilem5  16122  dvbsss  21513  perfdvf  21514  uhgra0  23415  umgra0  23431  usgra0  23461  eupa0  23767  ismgm  23979  dmadjrnb  25482  mbfmcst  26838  0rrv  26998  2wlkonot3v  30562  2spthonot3v  30563  clwwlknprop  30603
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