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Theorem nel0 4350
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 4343 . 2 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
2 nel0.1 . 2 ¬ 𝑥𝐴
31, 2mpgbir 1793 1 𝐴 = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1533  wcel 2098  c0 4322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-dif 3947  df-nul 4323
This theorem is referenced by:  iun0  5066  0iun  5067  0xp  5776  dm0  5923  cnv0  6147  fzouzdisj  13703  bj-ccinftydisj  36823  finxp0  37001  stoweidlem44  45570
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