Theorem rab0 3391

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 3367	. . . . 5 |- -. x e. (/)
2	1	intnanr 915	. . . 4 |- -. ( x e. (/) /\ ph )
3		equid 1677	. . . . 5 |- x = x
4	3	notnoti 634	. . . 4 |- -. -. x = x
5	2, 4	2false 690	. . 3 |- ( ( x e. (/) /\ ph ) <-> -. x = x )
6	5	abbii 2255	. 2 |- { x | ( x e. (/) /\ ph ) } = { x | -. x = x }
7		df-rab 2425	. 2 |- { x e. (/) | ph } = { x | ( x e. (/) /\ ph ) }
8		dfnul2 3365	. 2 |- (/) = { x | -. x = x }
9	6, 7, 8	3eqtr4i 2170	1 |- { x e. (/) | ph } = (/)

Syntax hints:   -. wn 3   /\ wa 103   = wceq 1331   e. wcel 1480   {cab 2125   {crab 2420   (/)c0 3363

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-dif 3073  df-nul 3364

This theorem is referenced by:  ssfirab  6822  sup00  6890