Theorem dm0 2542

Description: The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36.

Assertion

Ref	Expression
dm0	|- dom (/) = (/)

Proof of Theorem dm0

Step	Hyp	Ref	Expression
1		noel 1711	. . . . 5 |- -. <.x, y>. e. (/)
2	1	nex 779	. . . 4 |- -. E.y<.x, y>. e. (/)
3		cleqid 1102	. . . . 5 |- x = x
4		negb 79	. . . . 5 |- (x = x -> -. -. x = x)
5	3, 4	ax-mp 6	. . . 4 |- -. -. x = x
6		pm5.21 502	. . . 4 |- ((-. E.y<.x, y>. e. (/) /\ -. -. x = x) -> (E.y<.x, y>. e. (/) <-> -. x = x))
7	2, 5, 6	mp2an 520	. . 3 |- (E.y<.x, y>. e. (/) <-> -. x = x)
8	7	biabi 1181	. 2 |- {x | E.y<.x, y>. e. (/)} = {x | -. x = x}
9		dfdm3 2522	. 2 |- dom (/) = {x | E.y<.x, y>. e. (/)}
10		dfnul2 1709	. 2 |- (/) = {x | -. x = x}
11	8, 9, 10	3eqtr4 1126	1 |- dom (/) = (/)

Syntax hints:  -. wn 1   <-> wb 127  E.wex 678   = weq 797  {cab 1090   = wceq 1091   e. wcel 1092  (/)c0 1707  <.cop 1810  dom cdm 2410

This theorem is referenced by:  dmxpid 2553  rn0 2567  fn0 2739  f1o00 2823  tz7.44lem1 2965  tz7.44-2 2967  1stval 3089  infxpidmlem4 4936

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-nul 1708  df-br 2063  df-dm 2428

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