Theorem dm0 4567

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 3266	. 2 |- ( dom (/) = (/) <-> A. x -. x e. dom (/) )
2		noel 3255	. . . 4 |- -. <. x , y >. e. (/)
3	2	nex 1429	. . 3 |- -. E. y <. x , y >. e. (/)
4		vex 2604	. . . 4 |- x e. _V
5	4	eldm2 4551	. . 3 |- ( x e. dom (/) <-> E. y <. x , y >. e. (/) )
6	3, 5	mtbir 628	. 2 |- -. x e. dom (/)
7	1, 6	mpgbir 1382	1 |- dom (/) = (/)

Syntax hints:   -. wn 3   = wceq 1284   E.wex 1421   e. wcel 1433   (/)c0 3251   <.cop 3401   dom cdm 4363

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-un 2977  df-nul 3252  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-dm 4373

This theorem is referenced by:  rn0  4606  fn0  5038  f1o00  5181  rdg0  5997  frec0g  6006