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Theorem abn0r 3271
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 2044 . . 3  |-  ( x  e.  { x  | 
ph }  <->  ph )
21exbii 1512 . 2  |-  ( E. x  x  e.  {
x  |  ph }  <->  E. x ph )
3 nfab1 2196 . . 3  |-  F/_ x { x  |  ph }
43n0rf 3261 . 2  |-  ( E. x  x  e.  {
x  |  ph }  ->  { x  |  ph }  =/=  (/) )
52, 4sylbir 129 1  |-  ( E. x ph  ->  { x  |  ph }  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1397    e. wcel 1409   {cab 2042    =/= wne 2220   (/)c0 3252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-v 2576  df-dif 2948  df-nul 3253
This theorem is referenced by:  rabn0r  3272
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