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Theorem rab0 3792
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
StepHypRef Expression
1 equid 1777 . . . . 5  |-  x  =  x
2 noel 3774 . . . . . 6  |-  -.  x  e.  (/)
32intnanr 915 . . . . 5  |-  -.  (
x  e.  (/)  /\  ph )
41, 32th 239 . . . 4  |-  ( x  =  x  <->  -.  (
x  e.  (/)  /\  ph ) )
54con2bii 332 . . 3  |-  ( ( x  e.  (/)  /\  ph ) 
<->  -.  x  =  x )
65abbii 2577 . 2  |-  { x  |  ( x  e.  (/)  /\  ph ) }  =  { x  |  -.  x  =  x }
7 df-rab 2802 . 2  |-  { x  e.  (/)  |  ph }  =  { x  |  ( x  e.  (/)  /\  ph ) }
8 dfnul2 3772 . 2  |-  (/)  =  {
x  |  -.  x  =  x }
96, 7, 83eqtr4i 2482 1  |-  { x  e.  (/)  |  ph }  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1383    e. wcel 1804   {cab 2428   {crab 2797   (/)c0 3770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rab 2802  df-v 3097  df-dif 3464  df-nul 3771
This theorem is referenced by:  rabsnif  4084  supp0  6908  scott0  8307  psgnfval  16399  pmtrsn  16418  00lsp  17501  rrgval  17809  usgra0v  24243  vdgr0  24772
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