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Theorem bj-ru 33419
Description: Remove dependency on ax-13 2354 (and df-v 3385) from Russell's paradox ru 3630 expressed with primitive symbols and with a class variable 𝑉 (note that axsep2 4974 does require ax-8 2159 and ax-9 2166 since it requires df-clel 2793 and df-cleq 2790--- see bj-df-clel 33371 and bj-df-cleq 33376). Note the more economical use of bj-elissetv 33344 instead of isset 3393 to avoid use of df-v 3385. (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 33418 . 2 ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥𝑥}
2 bj-elissetv 33344 . 2 ({𝑥 ∣ ¬ 𝑥𝑥} ∈ 𝑉 → ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥𝑥})
31, 2mto 189 1 ¬ {𝑥 ∣ ¬ 𝑥𝑥} ∈ 𝑉
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1653  wex 1875  wcel 2157  {cab 2783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793
This theorem is referenced by: (None)
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