ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  abn0r Unicode version

Theorem abn0r 3326
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 2083 . . 3  |-  ( x  e.  { x  | 
ph }  <->  ph )
21exbii 1548 . 2  |-  ( E. x  x  e.  {
x  |  ph }  <->  E. x ph )
3 nfab1 2237 . . 3  |-  F/_ x { x  |  ph }
43n0rf 3314 . 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 1433    e. wcel 1445   {cab 2081    =/= wne 2262   (/)c0 3302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-v 2635  df-dif 3015  df-nul 3303
This theorem is referenced by:  rabn0r  3328
  Copyright terms: Public domain W3C validator