Theorem rab0 3418

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 3394	. . . . 5 |- -. x e. (/)
2	1	intnanr 916	. . . 4 |- -. ( x e. (/) /\ ph )
3		equid 1678	. . . . 5 |- x = x
4	3	notnoti 635	. . . 4 |- -. -. x = x
5	2, 4	2false 691	. . 3 |- ( ( x e. (/) /\ ph ) <-> -. x = x )
6	5	abbii 2270	. 2 |- { x | ( x e. (/) /\ ph ) } = { x | -. x = x }
7		df-rab 2441	. 2 |- { x e. (/) | ph } = { x | ( x e. (/) /\ ph ) }
8		dfnul2 3392	. 2 |- (/) = { x | -. x = x }
9	6, 7, 8	3eqtr4i 2185	1 |- { x e. (/) | ph } = (/)

Syntax hints:   -. wn 3   /\ wa 103   = wceq 1332   e. wcel 2125   {cab 2140   {crab 2436   (/)c0 3390

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-rab 2441  df-v 2711  df-dif 3100  df-nul 3391

This theorem is referenced by:  ssfirab  6867  sup00  6935