Theorem rab0 3955

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.) (Proof shortened by JJ, 14-Jul-2021.)

Assertion

Ref	Expression
rab0	|- { x e. (/) | ph } = (/)

Proof of Theorem rab0

Step	Hyp	Ref	Expression
1		df-rab 2921	. 2 |- { x e. (/) | ph } = { x | ( x e. (/) /\ ph ) }
2		ab0 3951	. . 3 |- ( { x | ( x e. (/) /\ ph ) } = (/) <-> A. x -. ( x e. (/) /\ ph ) )
3		noel 3919	. . . 4 |- -. x e. (/)
4	3	intnanr 961	. . 3 |- -. ( x e. (/) /\ ph )
5	2, 4	mpgbir 1726	. 2 |- { x | ( x e. (/) /\ ph ) } = (/)
6	1, 5	eqtri 2644	1 |- { x e. (/) | ph } = (/)

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  rabsnif  4258  supp0  7300  sup00  8370  scott0  8749  psgnfval  17920  pmtrsn  17939  00lsp  18981  rrgval  19287  uvtxa0  26294  vtxdg0e  26370  wwlksn  26729  wspthsn  26735  iswwlksnon  26740  iswspthsnon  26741  clwwlksn  26881