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Theorem neq0r 3345
Description: An inhabited class is nonempty. See n0rf 3343 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 3344 . 2  |-  ( E. x  x  e.  A  ->  A  =/=  (/) )
21neneqd 2304 1  |-  ( E. x  x  e.  A  ->  -.  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1314   E.wex 1451    e. wcel 1463   (/)c0 3331
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-v 2660  df-dif 3041  df-nul 3332
This theorem is referenced by:  exmidsssn  4093  fzn  9773
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