Theorem dm0 4577

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 3273	. 2 |- ( dom (/) = (/) <-> A. x -. x e. dom (/) )
2		noel 3262	. . . 4 |- -. <. x , y >. e. (/)
3	2	nex 1430	. . 3 |- -. E. y <. x , y >. e. (/)
4		vex 2605	. . . 4 |- x e. _V
5	4	eldm2 4561	. . 3 |- ( x e. dom (/) <-> E. y <. x , y >. e. (/) )
6	3, 5	mtbir 629	. 2 |- -. x e. dom (/)
7	1, 6	mpgbir 1383	1 |- dom (/) = (/)

Syntax hints:   -. wn 3   = wceq 1285   E.wex 1422   e. wcel 1434   (/)c0 3258   <.cop 3409   dom cdm 4371

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-dif 2976  df-un 2978  df-nul 3259  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-dm 4381

This theorem is referenced by:  rn0  4616  fn0  5049  f1o00  5192  rdg0  6036  frec0g  6046