Theorem rab0 3273

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 3255	. . . . 5 |- -. x e. (/)
2	1	intnanr 872	. . . 4 |- -. ( x e. (/) /\ ph )
3		equid 1629	. . . . 5 |- x = x
4	3	notnoti 606	. . . 4 |- -. -. x = x
5	2, 4	2false 649	. . 3 |- ( ( x e. (/) /\ ph ) <-> -. x = x )
6	5	abbii 2194	. 2 |- { x | ( x e. (/) /\ ph ) } = { x | -. x = x }
7		df-rab 2357	. 2 |- { x e. (/) | ph } = { x | ( x e. (/) /\ ph ) }
8		dfnul2 3253	. 2 |- (/) = { x | -. x = x }
9	6, 7, 8	3eqtr4i 2111	1 |- { x e. (/) | ph } = (/)

Syntax hints:   -. wn 3   /\ wa 102   = wceq 1284   e. wcel 1433   {cab 2067   {crab 2352   (/)c0 3251

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-dif 2975  df-nul 3252

This theorem is referenced by:  sup00  6416