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Theorem neq0r 3232
Description: An inhabited class is nonempty. See n0rf 3230 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
Assertion
Ref Expression
neq0r (∃𝑥 𝑥𝐴 → ¬ 𝐴 = ∅)
Distinct variable group:   𝑥,𝐴

Proof of Theorem neq0r
StepHypRef Expression
1 n0r 3231 . 2 (∃𝑥 𝑥𝐴𝐴 ≠ ∅)
21neneqd 2224 1 (∃𝑥 𝑥𝐴 → ¬ 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1243  wex 1381  wcel 1393  c0 3221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2556  df-dif 2917  df-nul 3222
This theorem is referenced by:  fzn  8835
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