Theorem dm0 4549

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 3239	. 2 |- ( dom (/) = (/) <-> A. x -. x e. dom (/) )
2		noel 3228	. . . 4 |- -. <. x , y >. e. (/)
3	2	nex 1389	. . 3 |- -. E. y <. x , y >. e. (/)
4		vex 2560	. . . 4 |- x e. _V
5	4	eldm2 4533	. . 3 |- ( x e. dom (/) <-> E. y <. x , y >. e. (/) )
6	3, 5	mtbir 596	. 2 |- -. x e. dom (/)
7	1, 6	mpgbir 1342	1 |- dom (/) = (/)

Syntax hints:   -. wn 3   = wceq 1243   E.wex 1381   e. wcel 1393   (/)c0 3224   <.cop 3378   dom cdm 4345

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-nul 3225  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-dm 4355

This theorem is referenced by:  rn0  4588  fn0  5018  f1o00  5161  rdg0  5974  frec0g  5983