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Theorem notm0 3378
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 3376 . 2 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
2 alnex 1475 . 2 (∀𝑥 ¬ 𝑥𝐴 ↔ ¬ ∃𝑥 𝑥𝐴)
31, 2bitr2i 184 1 (¬ ∃𝑥 𝑥𝐴𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 104  wal 1329   = wceq 1331  wex 1468  wcel 1480  c0 3358
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-nul 3359
This theorem is referenced by:  disjnim  3915  pwntru  4117  exmidn0m  4119  mapprc  6539  map0g  6575  ixpprc  6606  ixp0  6618  exmidfodomrlemim  7050  ntreq0  12290  blssioo  12703  pwtrufal  13181
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