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Theorem abn0r 3382
Description: Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
Assertion
Ref Expression
abn0r  |-  ( E. x ph  ->  { x  |  ph }  =/=  (/) )

Proof of Theorem abn0r
StepHypRef Expression
1 abid 2125 . . 3  |-  ( x  e.  { x  | 
ph }  <->  ph )
21exbii 1584 . 2  |-  ( E. x  x  e.  {
x  |  ph }  <->  E. x ph )
3 nfab1 2281 . . 3  |-  F/_ x { x  |  ph }
43n0rf 3370 . 2  |-  ( E. x  x  e.  {
x  |  ph }  ->  { x  |  ph }  =/=  (/) )
52, 4sylbir 134 1  |-  ( E. x ph  ->  { x  |  ph }  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1468    e. wcel 1480   {cab 2123    =/= wne 2306   (/)c0 3358
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-v 2683  df-dif 3068  df-nul 3359
This theorem is referenced by:  rabn0r  3384
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