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Theorem bj-nel0 32128
 Description: From the general negation of membership in 𝐴, infer that 𝐴 is the empty set. [Could shorten 0xp 5122?] (Contributed by BJ, 6-Oct-2018.)
Hypothesis
Ref Expression
bj-nel0.1 ¬ 𝑥𝐴
Assertion
Ref Expression
bj-nel0 𝐴 = ∅
Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-nel0
StepHypRef Expression
1 eq0 3888 . 2 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
2 bj-nel0.1 . 2 ¬ 𝑥𝐴
31, 2mpgbir 1717 1 𝐴 = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-nul 3875 This theorem is referenced by:  bj-ccinftydisj  32277
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