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Theorem neq0r 3297
Description: An inhabited class is nonempty. See n0rf 3295 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
Assertion
Ref Expression
neq0r  |-  ( E. x  x  e.  A  ->  -.  A  =  (/) )
Distinct variable group:    x, A

Proof of Theorem neq0r
StepHypRef Expression
1 n0r 3296 . 2  |-  ( E. x  x  e.  A  ->  A  =/=  (/) )
21neneqd 2276 1  |-  ( E. x  x  e.  A  ->  -.  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1289   E.wex 1426    e. wcel 1438   (/)c0 3286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-dif 3001  df-nul 3287
This theorem is referenced by:  fzn  9446
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