Theorem rab0 3805

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 1796	. . . . 5 |- x = x
2		noel 3787	. . . . . 6 |- -. x e. (/)
3	2	intnanr 913	. . . . 5 |- -. ( x e. (/) /\ ph )
4	1, 3	2th 239	. . . 4 |- ( x = x <-> -. ( x e. (/) /\ ph ) )
5	4	con2bii 330	. . 3 |- ( ( x e. (/) /\ ph ) <-> -. x = x )
6	5	abbii 2588	. 2 |- { x | ( x e. (/) /\ ph ) } = { x | -. x = x }
7		df-rab 2813	. 2 |- { x e. (/) | ph } = { x | ( x e. (/) /\ ph ) }
8		dfnul2 3785	. 2 |- (/) = { x | -. x = x }
9	6, 7, 8	3eqtr4i 2493	1 |- { x e. (/) | ph } = (/)

Syntax hints:   -. wn 3   /\ wa 367   = wceq 1398   e. wcel 1823   {cab 2439   {crab 2808   (/)c0 3783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2813  df-v 3108  df-dif 3464  df-nul 3784

This theorem is referenced by:  rabsnif  4085  supp0  6896  scott0  8295  psgnfval  16724  pmtrsn  16743  00lsp  17822  rrgval  18130  usgra0v  24573  vdgr0  25102