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Theorem abn0r 3459
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 2175 . . 3  |-  ( x  e.  { x  | 
ph }  <->  ph )
21exbii 1615 . 2  |-  ( E. x  x  e.  {
x  |  ph }  <->  E. x ph )
3 nfab1 2331 . . 3  |-  F/_ x { x  |  ph }
43n0rf 3447 . 2  |-  ( E. x  x  e.  {
x  |  ph }  ->  { x  |  ph }  =/=  (/) )
52, 4sylbir 135 1  |-  ( E. x ph  ->  { x  |  ph }  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1502    e. wcel 2158   {cab 2173    =/= wne 2357   (/)c0 3434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-v 2751  df-dif 3143  df-nul 3435
This theorem is referenced by:  rabn0r  3461
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