Theorem rab0 3541

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		noel 3516	. . . . 5 |- -. x e. (/)
2	1	intnanr 938	. . . 4 |- -. ( x e. (/) /\ ph )
3		equid 1749	. . . . 5 |- x = x
4	3	notnoti 650	. . . 4 |- -. -. x = x
5	2, 4	2false 709	. . 3 |- ( ( x e. (/) /\ ph ) <-> -. x = x )
6	5	abbii 2350	. 2 |- { x | ( x e. (/) /\ ph ) } = { x | -. x = x }
7		df-rab 2531	. 2 |- { x e. (/) | ph } = { x | ( x e. (/) /\ ph ) }
8		dfnul2 3514	. 2 |- (/) = { x | -. x = x }
9	6, 7, 8	3eqtr4i 2265	1 |- { x e. (/) | ph } = (/)

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1398   e. wcel 2205   {cab 2220   {crab 2526   (/)c0 3512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rab 2531  df-v 2817  df-dif 3216  df-nul 3513

This theorem is referenced by:  if0ab  3627  supp0  6451  ssfirab  7210  sup00  7307  vtxdgfi0e  16416  clwwlk0on0  16552