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Theorem dm0 4892
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 3469 . 2  |-  ( dom  (/)  =  (/)  <->  A. x  -.  x  e.  dom  (/) )
2 noel 3459 . . . 4  |-  -.  <. x ,  y >.  e.  (/)
32nex 1542 . . 3  |-  -.  E. y <. x ,  y
>.  e.  (/)
4 vex 2791 . . . 4  |-  x  e. 
_V
54eldm2 4877 . . 3  |-  ( x  e.  dom  (/)  <->  E. y <. x ,  y >.  e.  (/) )
63, 5mtbir 290 . 2  |-  -.  x  e.  dom  (/)
71, 6mpgbir 1537 1  |-  dom  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3   E.wex 1528    = wceq 1623    e. wcel 1684   (/)c0 3455   <.cop 3643   dom cdm 4689
This theorem is referenced by:  dmxpid  4898  rn0  4936  dmxpss  5107  fn0  5363  f1o00  5508  fv01  5559  1stval  6124  tz7.44lem1  6418  tz7.44-2  6420  tz7.44-3  6421  oicl  7244  oif  7245  strlemor0  13234  dvbsss  19252  perfdvf  19253  ismgm  20987  dmadjrnb  22486  mbfmcst  23564  0rrv  23654  umgra0  23877  eupa0  23898  0alg  25756  0ded  25757  0catOLD  25758  symgsssg  27408  symgfisg  27409  psgnunilem5  27417  usgra0  28116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-dm 4699
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