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Theorem nel0 4199
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 4196 . 2 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
2 nel0.1 . 2 ¬ 𝑥𝐴
31, 2mpgbir 1762 1 𝐴 = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1507  wcel 2050  c0 4180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-11 2093  ax-12 2106  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-dif 3834  df-nul 4181
This theorem is referenced by:  iun0  4852  0iun  4853  0xp  5500  dm0  5638  cnv0  5841  fzouzdisj  12891  bj-ccinftydisj  33964  finxp0  34113  stoweidlem44  41761
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