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Theorem bj-ru 32126
 Description: Remove dependency on ax-13 2234 (and df-v 3175) from Russell's paradox ru 3401 expressed with primitive symbols and with a class variable 𝑉 (note that axsep2 4710 does require ax-8 1979 and ax-9 1986 since it requires df-clel 2606 and df-cleq 2603--- see bj-df-clel 32081 and bj-df-cleq 32085). Note the more economical use of bj-elissetv 32055 instead of isset 3180 to avoid use of df-v 3175. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ru ¬ {𝑥 ∣ ¬ 𝑥𝑥} ∈ 𝑉

Proof of Theorem bj-ru
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 bj-ru1 32125 . 2 ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥𝑥}
2 bj-elissetv 32055 . 2 ({𝑥 ∣ ¬ 𝑥𝑥} ∈ 𝑉 → ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥𝑥})
31, 2mto 187 1 ¬ {𝑥 ∣ ¬ 𝑥𝑥} ∈ 𝑉
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596 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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  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 This theorem is referenced by: (None)
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