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Theorem bj-ru1 34373
Description: A version of Russell's paradox ru 3722 (see also bj-ru 34374). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ru1 ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥𝑥}
Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-ru1
StepHypRef Expression
1 bj-ru0 34372 . . 3 ¬ ∀𝑥(𝑥𝑦 ↔ ¬ 𝑥𝑥)
2 abeq2 2925 . . 3 (𝑦 = {𝑥 ∣ ¬ 𝑥𝑥} ↔ ∀𝑥(𝑥𝑦 ↔ ¬ 𝑥𝑥))
31, 2mtbir 326 . 2 ¬ 𝑦 = {𝑥 ∣ ¬ 𝑥𝑥}
43nex 1802 1 ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥𝑥}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wal 1536   = wceq 1538  wex 1781  {cab 2779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873
This theorem is referenced by:  bj-ru  34374
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