Theorem rab0 3443

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 3418	. . . . 5 |- -. x e. (/)
2	1	intnanr 925	. . . 4 |- -. ( x e. (/) /\ ph )
3		equid 1694	. . . . 5 |- x = x
4	3	notnoti 640	. . . 4 |- -. -. x = x
5	2, 4	2false 696	. . 3 |- ( ( x e. (/) /\ ph ) <-> -. x = x )
6	5	abbii 2286	. 2 |- { x | ( x e. (/) /\ ph ) } = { x | -. x = x }
7		df-rab 2457	. 2 |- { x e. (/) | ph } = { x | ( x e. (/) /\ ph ) }
8		dfnul2 3416	. 2 |- (/) = { x | -. x = x }
9	6, 7, 8	3eqtr4i 2201	1 |- { x e. (/) | ph } = (/)

Syntax hints:   -. wn 3   /\ wa 103   = wceq 1348   e. wcel 2141   {cab 2156   {crab 2452   (/)c0 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-dif 3123  df-nul 3415

This theorem is referenced by:  ssfirab  6911  sup00  6980