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Theorem rab0 3488
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 1662 . . . . 5  |-  x  =  x
2 noel 3472 . . . . . 6  |-  -.  x  e.  (/)
32intnanr 881 . . . . 5  |-  -.  (
x  e.  (/)  /\  ph )
41, 32th 230 . . . 4  |-  ( x  =  x  <->  -.  (
x  e.  (/)  /\  ph ) )
54con2bii 322 . . 3  |-  ( ( x  e.  (/)  /\  ph ) 
<->  -.  x  =  x )
65abbii 2408 . 2  |-  { x  |  ( x  e.  (/)  /\  ph ) }  =  { x  |  -.  x  =  x }
7 df-rab 2565 . 2  |-  { x  e.  (/)  |  ph }  =  { x  |  ( x  e.  (/)  /\  ph ) }
8 dfnul2 3470 . 2  |-  (/)  =  {
x  |  -.  x  =  x }
96, 7, 83eqtr4i 2326 1  |-  { x  e.  (/)  |  ph }  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   {crab 2560   (/)c0 3468
This theorem is referenced by:  scott0  7572  00lsp  15754  rrgval  16044  vdgr0  23907  psgnfval  27526  usgra0v  28251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-nul 3469
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