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Theorem rab0 3805
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 1796 . . . . 5  |-  x  =  x
2 noel 3787 . . . . . 6  |-  -.  x  e.  (/)
32intnanr 913 . . . . 5  |-  -.  (
x  e.  (/)  /\  ph )
41, 32th 239 . . . 4  |-  ( x  =  x  <->  -.  (
x  e.  (/)  /\  ph ) )
54con2bii 330 . . 3  |-  ( ( x  e.  (/)  /\  ph ) 
<->  -.  x  =  x )
65abbii 2588 . 2  |-  { x  |  ( x  e.  (/)  /\  ph ) }  =  { x  |  -.  x  =  x }
7 df-rab 2813 . 2  |-  { x  e.  (/)  |  ph }  =  { x  |  ( x  e.  (/)  /\  ph ) }
8 dfnul2 3785 . 2  |-  (/)  =  {
x  |  -.  x  =  x }
96, 7, 83eqtr4i 2493 1  |-  { x  e.  (/)  |  ph }  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 367    = wceq 1398    e. wcel 1823   {cab 2439   {crab 2808   (/)c0 3783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2813  df-v 3108  df-dif 3464  df-nul 3784
This theorem is referenced by:  rabsnif  4085  supp0  6896  scott0  8295  psgnfval  16724  pmtrsn  16743  00lsp  17822  rrgval  18130  usgra0v  24573  vdgr0  25102
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