Theorem rab0 1718

Description: Any restricted class abstraction restricted to the empty set is empty.

Assertion

Ref	Expression
rab0	|- {x e. (/) | ph} = (/)

Proof of Theorem rab0

Step	Hyp	Ref	Expression
1		noel 1711	. . . 4 |- -. x e. (/)
2	1	intnanr 517	. . 3 |- -. (x e. (/) /\ ph)
3	2	nex 779	. 2 |- -. E.x(x e. (/) /\ ph)
4		rabn0 1716	. . . 4 |- (-. {x e. (/) | ph} = (/) <-> E.x e. (/) ph)
5		df-rex 1206	. . . 4 |- (E.x e. (/) ph <-> E.x(x e. (/) /\ ph))
6	4, 5	bitr 151	. . 3 |- (-. {x e. (/) | ph} = (/) <-> E.x(x e. (/) /\ ph))
7	6	bicon1i 193	. 2 |- (-. E.x(x e. (/) /\ ph) <-> {x e. (/) | ph} = (/))
8	3, 7	mpbi 164	1 |- {x e. (/) | ph} = (/)

Syntax hints:  -. wn 1   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  E.wrex 1202  {crab 1204  (/)c0 1707

This theorem is referenced by:  scott0 3542

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-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-nul 1708

metamath.org