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Theorem dm0 5046
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 3606 . 2  |-  ( dom  (/)  =  (/)  <->  A. x  -.  x  e.  dom  (/) )
2 noel 3596 . . . 4  |-  -.  <. x ,  y >.  e.  (/)
32nex 1561 . . 3  |-  -.  E. y <. x ,  y
>.  e.  (/)
4 vex 2923 . . . 4  |-  x  e. 
_V
54eldm2 5031 . . 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 3592   <.cop 3781   dom cdm 4841
This theorem is referenced by:  dmxpid  5052  rn0  5090  dmxpss  5263  fn0  5527  f1o00  5673  fv01  5726  1stval  6314  bropopvvv  6389  tz7.44lem1  6626  tz7.44-2  6628  tz7.44-3  6629  oicl  7458  oif  7459  strlemor0  13514  dvbsss  19746  perfdvf  19747  uhgra0  21301  umgra0  21317  usgra0  21347  eupa0  21653  ismgm  21865  dmadjrnb  23366  mbfmcst  24566  0rrv  24666  symgsssg  27280  symgfisg  27281  psgnunilem5  27289  swrd0  28006  2wlkonot3v  28076  2spthonot3v  28077
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 2389
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 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-dm 4851
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