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Theorem rab0 3359
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 noel 3335 . . . . 5  |-  -.  x  e.  (/)
21intnanr 898 . . . 4  |-  -.  (
x  e.  (/)  /\  ph )
3 equid 1660 . . . . 5  |-  x  =  x
43notnoti 617 . . . 4  |-  -.  -.  x  =  x
52, 42false 673 . . 3  |-  ( ( x  e.  (/)  /\  ph ) 
<->  -.  x  =  x )
65abbii 2231 . 2  |-  { x  |  ( x  e.  (/)  /\  ph ) }  =  { x  |  -.  x  =  x }
7 df-rab 2400 . 2  |-  { x  e.  (/)  |  ph }  =  { x  |  ( x  e.  (/)  /\  ph ) }
8 dfnul2 3333 . 2  |-  (/)  =  {
x  |  -.  x  =  x }
96, 7, 83eqtr4i 2146 1  |-  { x  e.  (/)  |  ph }  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    = wceq 1314    e. wcel 1463   {cab 2101   {crab 2395   (/)c0 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rab 2400  df-v 2660  df-dif 3041  df-nul 3332
This theorem is referenced by:  ssfirab  6788  sup00  6856
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