Theorem dm0 4880

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.

Step	Hyp	Ref	Expression
1		eq0 3469	. 2 |- ( dom (/) = (/) <-> A. x -. x e. dom (/) )
2		noel 3454	. . . 4 |- -. <. x , y >. e. (/)
3	2	nex 1514	. . 3 |- -. E. y <. x , y >. e. (/)
4		vex 2766	. . . 4 |- x e. _V
5	4	eldm2 4864	. . 3 |- ( x e. dom (/) <-> E. y <. x , y >. e. (/) )
6	3, 5	mtbir 672	. 2 |- -. x e. dom (/)
7	1, 6	mpgbir 1467	1 |- dom (/) = (/)

Syntax hints:   -. wn 3   = wceq 1364   E.wex 1506   e. wcel 2167   (/)c0 3450   <.cop 3625   dom cdm 4663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-un 3161  df-nul 3451  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-dm 4673

This theorem is referenced by:  rn0  4922  sqxpeq0  5093  fn0  5377  f0dom0  5451  f1o00  5539  rdg0  6445  frec0g  6455  ennnfonelemj0  12618  ennnfonelem1  12624  ennnfonelemkh  12629  ennnfonelemhf1o  12630