Theorem rab0 2293

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 2284	. . . 4 |- -. x e. (/)
2	1	intnanr 692	. . 3 |- -. (x e. (/) /\ ph)
3	2	nex 1101	. 2 |- -. E.x(x e. (/) /\ ph)
4		rabn0 2292	. . . 4 |- ({x e. (/) | ph} =/= (/) <-> E.x e. (/) ph)
5		df-rex 1650	. . . 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 1617	. 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 956   e. wcel 958  E.wex 980   =/= wne 1585  E.wrex 1646  {crab 1648  (/)c0 2280

This theorem is referenced by:  scott0 4717

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-nul 2281

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