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Theorem notm0 3455
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 3453 . 2 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
2 alnex 1509 . 2 (∀𝑥 ¬ 𝑥𝐴 ↔ ¬ ∃𝑥 𝑥𝐴)
31, 2bitr2i 185 1 (¬ ∃𝑥 𝑥𝐴𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 105  wal 1361   = wceq 1363  wex 1502  wcel 2158  c0 3434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-dif 3143  df-nul 3435
This theorem is referenced by:  disjnim  4006  pwntru  4211  exmidn0m  4213  mapprc  6666  map0g  6702  ixpprc  6733  ixp0  6745  exmidfodomrlemim  7214  ntreq0  13928  blssioo  14341  pwtrufal  15044
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