Theorem rab0 2289

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 2280	. . . 4 |- -. x e. (/)
2	1	intnanr 691	. . 3 |- -. (x e. (/) /\ ph)
3	2	nex 1099	. 2 |- -. E.x(x e. (/) /\ ph)
4		rabn0 2288	. . . 4 |- ({x e. (/) | ph} =/= (/) <-> E.x e. (/) ph)
5		df-rex 1647	. . . 4 |- (E.x e. (/) ph <-> E.x(x e. (/) /\ ph))
6	4, 5	bitr 173	. . 3 |- ({x e. (/) | ph} =/= (/) <-> E.x(x e. (/) /\ ph))
7	6	necon1bbii 1614	. 2 |- (-. E.x(x e. (/) /\ ph) <-> {x e. (/) | ph} = (/))
8	3, 7	mpbi 189	1 |- {x e. (/) | ph} = (/)

Syntax hints:  -. wn 2   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978   =/= wne 1582  E.wrex 1643  {crab 1645  (/)c0 2276

This theorem is referenced by:  scott0 4697

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-rab 1649  df-v 1808  df-dif 2045  df-nul 2277

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