Theorem rab0 2894

Description: Any restricted class abstraction restricted to the empty set is empty. (The proof was shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem rab0

Step	Hyp	Ref	Expression
1		equid 1484	. . . . 5 |- x = x
2		noel 2879	. . . . . 6 |- -. x e. (/)
3	2	intnanr 756	. . . . 5 |- -. (x e. (/) /\ ph)
4	1, 3	2th 786	. . . 4 |- (x = x <-> -. (x e. (/) /\ ph))
5	4	con2bii 238	. . 3 |- ((x e. (/) /\ ph) <-> -. x = x)
6	5	abbii 2006	. 2 |- {x | (x e. (/) /\ ph)} = {x | -. x = x}
7		df-rab 2112	. 2 |- {x e. (/) | ph} = {x | (x e. (/) /\ ph)}
8		dfnul2 2877	. 2 |- (/) = {x | -. x = x}
9	6, 7, 8	3eqtr4i 1921	1 |- {x e. (/) | ph} = (/)

Syntax hints:  -. wn 2   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  {crab 2108  (/)c0 2875

This theorem is referenced by:  scott0 5847  gid0 9338

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-rab 2112  df-v 2294  df-dif 2597  df-nul 2876

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