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Theorem dm0 5050
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 3610 . 2  |-  ( dom  (/)  =  (/)  <->  A. x  -.  x  e.  dom  (/) )
2 noel 3600 . . . 4  |-  -.  <. x ,  y >.  e.  (/)
32nex 1561 . . 3  |-  -.  E. y <. x ,  y
>.  e.  (/)
4 vex 2927 . . . 4  |-  x  e. 
_V
54eldm2 5035 . . 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 1721   (/)c0 3596   <.cop 3785   dom cdm 4845
This theorem is referenced by:  dmxpid  5056  rn0  5094  dmxpss  5267  fn0  5531  f1o00  5677  fv01  5730  1stval  6318  bropopvvv  6393  tz7.44lem1  6630  tz7.44-2  6632  tz7.44-3  6633  oicl  7462  oif  7463  strlemor0  13518  dvbsss  19750  perfdvf  19751  uhgra0  21305  umgra0  21321  usgra0  21351  eupa0  21657  ismgm  21869  dmadjrnb  23370  mbfmcst  24570  0rrv  24670  symgsssg  27284  symgfisg  27285  psgnunilem5  27293  swrd0  28010  2wlkonot3v  28080  2spthonot3v  28081
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
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 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-dm 4855
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