Theorem dm0 4638

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 3299	. 2 |- ( dom (/) = (/) <-> A. x -. x e. dom (/) )
2		noel 3288	. . . 4 |- -. <. x , y >. e. (/)
3	2	nex 1434	. . 3 |- -. E. y <. x , y >. e. (/)
4		vex 2622	. . . 4 |- x e. _V
5	4	eldm2 4622	. . 3 |- ( x e. dom (/) <-> E. y <. x , y >. e. (/) )
6	3, 5	mtbir 631	. 2 |- -. x e. dom (/)
7	1, 6	mpgbir 1387	1 |- dom (/) = (/)

Syntax hints:   -. wn 3   = wceq 1289   E.wex 1426   e. wcel 1438   (/)c0 3284   <.cop 3444   dom cdm 4428

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-un 3001  df-nul 3285  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-dm 4438

This theorem is referenced by:  rn0  4677  fn0  5119  f0dom0  5188  f1o00  5272  rdg0  6134  frec0g  6144