Theorem rab0 3759

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 1731	. . . . 5 |- x = x
2		noel 3742	. . . . . 6 |- -. x e. (/)
3	2	intnanr 906	. . . . 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 2585	. 2 |- { x | ( x e. (/) /\ ph ) } = { x | -. x = x }
7		df-rab 2804	. 2 |- { x e. (/) | ph } = { x | ( x e. (/) /\ ph ) }
8		dfnul2 3740	. 2 |- (/) = { x | -. x = x }
9	6, 7, 8	3eqtr4i 2490	1 |- { x e. (/) | ph } = (/)

Syntax hints:   -. wn 3   /\ wa 369   = wceq 1370   e. wcel 1758   {cab 2436   {crab 2799   (/)c0 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rab 2804  df-v 3073  df-dif 3432  df-nul 3739

This theorem is referenced by:  rabsnif  4045  supp0  6798  scott0  8197  psgnfval  16117  pmtrsn  16136  00lsp  17177  rrgval  17473  usgra0v  23435  vdgr0  23715