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Theorem rab0 3806
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 1740 . . . . 5  |-  x  =  x
2 noel 3789 . . . . . 6  |-  -.  x  e.  (/)
32intnanr 913 . . . . 5  |-  -.  (
x  e.  (/)  /\  ph )
41, 32th 239 . . . 4  |-  ( x  =  x  <->  -.  (
x  e.  (/)  /\  ph ) )
54con2bii 332 . . 3  |-  ( ( x  e.  (/)  /\  ph ) 
<->  -.  x  =  x )
65abbii 2601 . 2  |-  { x  |  ( x  e.  (/)  /\  ph ) }  =  { x  |  -.  x  =  x }
7 df-rab 2823 . 2  |-  { x  e.  (/)  |  ph }  =  { x  |  ( x  e.  (/)  /\  ph ) }
8 dfnul2 3787 . 2  |-  (/)  =  {
x  |  -.  x  =  x }
96, 7, 83eqtr4i 2506 1  |-  { x  e.  (/)  |  ph }  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   {crab 2818   (/)c0 3785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-dif 3479  df-nul 3786
This theorem is referenced by:  rabsnif  4096  supp0  6903  scott0  8300  psgnfval  16321  pmtrsn  16340  00lsp  17410  rrgval  17706  usgra0v  24047  vdgr0  24576
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