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Theorem rab0 3591
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 1683 . . . . 5  |-  x  =  x
2 noel 3575 . . . . . 6  |-  -.  x  e.  (/)
32intnanr 882 . . . . 5  |-  -.  (
x  e.  (/)  /\  ph )
41, 32th 231 . . . 4  |-  ( x  =  x  <->  -.  (
x  e.  (/)  /\  ph ) )
54con2bii 323 . . 3  |-  ( ( x  e.  (/)  /\  ph ) 
<->  -.  x  =  x )
65abbii 2499 . 2  |-  { x  |  ( x  e.  (/)  /\  ph ) }  =  { x  |  -.  x  =  x }
7 df-rab 2658 . 2  |-  { x  e.  (/)  |  ph }  =  { x  |  ( x  e.  (/)  /\  ph ) }
8 dfnul2 3573 . 2  |-  (/)  =  {
x  |  -.  x  =  x }
96, 7, 83eqtr4i 2417 1  |-  { x  e.  (/)  |  ph }  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2373   {crab 2653   (/)c0 3571
This theorem is referenced by:  rabrsn  3817  scott0  7743  00lsp  15984  rrgval  16274  usgra0v  21258  vdgr0  21519  psgnfval  27092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rab 2658  df-v 2901  df-dif 3266  df-nul 3572
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