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Theorem bj-ru 34158
 Description: Remove dependency on ax-13 2385 (and df-v 3502) from Russell's paradox ru 3775 expressed with primitive symbols and with a class variable 𝑉. Note the more economical use of bj-elissetv 34094 instead of isset 3512 to avoid use of df-v 3502. (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 34157 . 2 ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥𝑥}
2 bj-elissetv 34094 . 2 ({𝑥 ∣ ¬ 𝑥𝑥} ∈ 𝑉 → ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥𝑥})
31, 2mto 198 1 ¬ {𝑥 ∣ ¬ 𝑥𝑥} ∈ 𝑉
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1530  ∃wex 1773   ∈ wcel 2107  {cab 2804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898 This theorem is referenced by: (None)
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