Theorem rab0 3806

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 1740	. . . . 5 |- x = x
2		noel 3789	. . . . . 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 332	. . 3 |- ( ( x e. (/) /\ ph ) <-> -. x = x )
6	5	abbii 2601	. 2 |- { x | ( x e. (/) /\ ph ) } = { x | -. x = x }
7		df-rab 2823	. 2 |- { x e. (/) | ph } = { x | ( x e. (/) /\ ph ) }
8		dfnul2 3787	. 2 |- (/) = { x | -. x = x }
9	6, 7, 8	3eqtr4i 2506	1 |- { x e. (/) | ph } = (/)

Syntax hints:   -. wn 3   /\ wa 369   = wceq 1379   e. wcel 1767   {cab 2452   {crab 2818   (/)c0 3785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-dif 3479  df-nul 3786

This theorem is referenced by:  rabsnif  4096  supp0  6903  scott0  8300  psgnfval  16321  pmtrsn  16340  00lsp  17410  rrgval  17706  usgra0v  24047  vdgr0  24576