Theorem rab0OLD 2895

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

Assertion

Ref	Expression
rab0OLD	|- {x e. (/) | ph} = (/)

Proof of Theorem rab0OLD

Step	Hyp	Ref	Expression
1		noel 2879	. . . 4 |- -. x e. (/)
2	1	intnanr 756	. . 3 |- -. (x e. (/) /\ ph)
3	2	nex 1456	. 2 |- -. E.x(x e. (/) /\ ph)
4		rabn0 2893	. . . 4 |- ({x e. (/) | ph} =/= (/) <-> E.x e. (/) ph)
5		df-rex 2110	. . . 4 |- (E.x e. (/) ph <-> E.x(x e. (/) /\ ph))
6	4, 5	bitri 190	. . 3 |- ({x e. (/) | ph} =/= (/) <-> E.x(x e. (/) /\ ph))
7	6	necon1bbii 2060	. 2 |- (-. E.x(x e. (/) /\ ph) <-> {x e. (/) | ph} = (/))
8	3, 7	mpbi 206	1 |- {x e. (/) | ph} = (/)

Syntax hints:  -. wn 2   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  E.wrex 2106  {crab 2108  (/)c0 2875

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-nul 2876

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