Theorem rab0 3792

Description: Any restricted class abstraction restricted to the empty set is empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem rab0

Step	Hyp	Ref	Expression
1		equid 1777	. . . . 5 |- x = x
2		noel 3774	. . . . . 6 |- -. x e. (/)
3	2	intnanr 915	. . . . 5 |- -. ( x e. (/) /\ ph )
4	1, 3	2th 239	. . . 4 |- ( x = x <-> -. ( x e. (/) /\ ph ) )
5	4	con2bii 332	. . 3 |- ( ( x e. (/) /\ ph ) <-> -. x = x )
6	5	abbii 2577	. 2 |- { x | ( x e. (/) /\ ph ) } = { x | -. x = x }
7		df-rab 2802	. 2 |- { x e. (/) | ph } = { x | ( x e. (/) /\ ph ) }
8		dfnul2 3772	. 2 |- (/) = { x | -. x = x }
9	6, 7, 8	3eqtr4i 2482	1 |- { x e. (/) | ph } = (/)

Syntax hints:   -. wn 3   /\ wa 369   = wceq 1383   e. wcel 1804   {cab 2428   {crab 2797   (/)c0 3770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rab 2802  df-v 3097  df-dif 3464  df-nul 3771

This theorem is referenced by:  rabsnif  4084  supp0  6908  scott0  8307  psgnfval  16399  pmtrsn  16418  00lsp  17501  rrgval  17809  usgra0v  24243  vdgr0  24772