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Theorem nel0 4351
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 4344 . 2 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
2 nel0.1 . 2 ¬ 𝑥𝐴
31, 2mpgbir 1802 1 𝐴 = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  c0 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-dif 3952  df-nul 4324
This theorem is referenced by:  iun0  5066  0iun  5067  0xp  5775  dm0  5921  cnv0  6141  fzouzdisj  13668  bj-ccinftydisj  36094  finxp0  36272  stoweidlem44  44760
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