Theorem rab0 3489

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 3464	. . . . 5 |- -. x e. (/)
2	1	intnanr 932	. . . 4 |- -. ( x e. (/) /\ ph )
3		equid 1724	. . . . 5 |- x = x
4	3	notnoti 646	. . . 4 |- -. -. x = x
5	2, 4	2false 703	. . 3 |- ( ( x e. (/) /\ ph ) <-> -. x = x )
6	5	abbii 2321	. 2 |- { x | ( x e. (/) /\ ph ) } = { x | -. x = x }
7		df-rab 2493	. 2 |- { x e. (/) | ph } = { x | ( x e. (/) /\ ph ) }
8		dfnul2 3462	. 2 |- (/) = { x | -. x = x }
9	6, 7, 8	3eqtr4i 2236	1 |- { x e. (/) | ph } = (/)

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1373   e. wcel 2176   {cab 2191   {crab 2488   (/)c0 3460

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-v 2774  df-dif 3168  df-nul 3461

This theorem is referenced by:  ssfirab  7033  sup00  7105