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Theorem dm0 4845
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
StepHypRef Expression
1 eq0 3411 . 2  |-  ( dom  (/)  =  (/)  <->  A. x  -.  x  e.  dom  (/) )
2 noel 3401 . . . 4  |-  -.  <. x ,  y >.  e.  (/)
32nex 1587 . . 3  |-  -.  E. y <. x ,  y
>.  e.  (/)
4 vex 2743 . . . 4  |-  x  e. 
_V
54eldm2 4830 . . 3  |-  ( x  e.  dom  (/)  <->  E. y <. x ,  y >.  e.  (/) )
63, 5mtbir 292 . 2  |-  -.  x  e.  dom  (/)
71, 6mpgbir 1544 1  |-  dom  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 5   E.wex 1537    = wceq 1619    e. wcel 1621   (/)c0 3397   <.cop 3584   dom cdm 4626
This theorem is referenced by:  dmxpid  4851  rn0  4889  dmxpss  5060  fn0  5266  f1o00  5411  fv01  5458  1stval  6023  tz7.44lem1  6351  tz7.44-2  6353  tz7.44-3  6354  oicl  7177  oif  7178  strlemor0  13161  dvbsss  19179  perfdvf  19180  ismgm  20912  dmadjrnb  22411  umgra0  23214  eupa0  23235  0alg  25088  0ded  25089  0catOLD  25090  symgsssg  26740  symgfisg  26741  psgnunilem5  26749
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-br 3964  df-dm 4644
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