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Theorem bj-ru 33059
Description: Remove dependency on ax-13 2282 (and df-v 3233) from Russell's paradox ru 3467 expressed with primitive symbols and with a class variable 𝑉 (note that axsep2 4815 does require ax-8 2032 and ax-9 2039 since it requires df-clel 2647 and df-cleq 2644--- see bj-df-clel 33013 and bj-df-cleq 33018). Note the more economical use of bj-elissetv 32986 instead of isset 3238 to avoid use of df-v 3233. (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 33058 . 2 ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥𝑥}
2 bj-elissetv 32986 . 2 ({𝑥 ∣ ¬ 𝑥𝑥} ∈ 𝑉 → ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥𝑥})
31, 2mto 188 1 ¬ {𝑥 ∣ ¬ 𝑥𝑥} ∈ 𝑉
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1523  wex 1744  wcel 2030  {cab 2637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647
This theorem is referenced by: (None)
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