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Theorem notm0 3286
Description: A class is not inhabited if and only if it is empty. (Contributed by Jim Kingdon, 1-Jul-2022.)
Assertion
Ref Expression
notm0 (¬ ∃𝑥 𝑥𝐴𝐴 = ∅)
Distinct variable group:   𝑥,𝐴

Proof of Theorem notm0
StepHypRef Expression
1 eq0 3284 . 2 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
2 alnex 1429 . 2 (∀𝑥 ¬ 𝑥𝐴 ↔ ¬ ∃𝑥 𝑥𝐴)
31, 2bitr2i 183 1 (¬ ∃𝑥 𝑥𝐴𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 103  wal 1283   = wceq 1285  wex 1422  wcel 1434  c0 3269
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-dif 2986  df-nul 3270
This theorem is referenced by:  mapprc  6337  map0g  6373  exmidfodomrlemim  6728
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