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Theorem nel0 4353
Description: From the general negation of membership in 𝐴, infer that 𝐴 is the empty set. (Contributed by BJ, 6-Oct-2018.)
Hypothesis
Ref Expression
nel0.1 ¬ 𝑥𝐴
Assertion
Ref Expression
nel0 𝐴 = ∅
Distinct variable group:   𝑥,𝐴

Proof of Theorem nel0
StepHypRef Expression
1 eq0 4349 . 2 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
2 nel0.1 . 2 ¬ 𝑥𝐴
31, 2mpgbir 1798 1 𝐴 = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1539  wcel 2107  c0 4332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-dif 3953  df-nul 4333
This theorem is referenced by:  iun0  5061  0iun  5062  0xp  5783  dm0  5930  cnv0  6159  fzouzdisj  13736  bj-ccinftydisj  37215  finxp0  37393  stoweidlem44  46064
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