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Theorem dm0 4823
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 3432 . 2  |-  ( dom  (/)  =  (/)  <->  A. x  -.  x  e.  dom  (/) )
2 noel 3418 . . . 4  |-  -.  <. x ,  y >.  e.  (/)
32nex 1493 . . 3  |-  -.  E. y <. x ,  y
>.  e.  (/)
4 vex 2733 . . . 4  |-  x  e. 
_V
54eldm2 4807 . . 3  |-  ( x  e.  dom  (/)  <->  E. y <. x ,  y >.  e.  (/) )
63, 5mtbir 666 . 2  |-  -.  x  e.  dom  (/)
71, 6mpgbir 1446 1  |-  dom  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1348   E.wex 1485    e. wcel 2141   (/)c0 3414   <.cop 3584   dom cdm 4609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-nul 3415  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-dm 4619
This theorem is referenced by:  rn0  4865  sqxpeq0  5032  fn0  5315  f0dom0  5389  f1o00  5475  rdg0  6363  frec0g  6373  ennnfonelemj0  12343  ennnfonelem1  12349  ennnfonelemkh  12354  ennnfonelemhf1o  12355
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