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Theorem bj-ru1 33361
Description: A version of Russell's paradox ru 3595 (see also bj-ru 33362). Note the more economical use of bj-abeq2 33203 instead of abeq2 2875 to avoid dependency on ax-13 2352. (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 33360 . . 3 ¬ ∀𝑥(𝑥𝑦 ↔ ¬ 𝑥𝑥)
2 bj-abeq2 33203 . . 3 (𝑦 = {𝑥 ∣ ¬ 𝑥𝑥} ↔ ∀𝑥(𝑥𝑦 ↔ ¬ 𝑥𝑥))
31, 2mtbir 314 . 2 ¬ 𝑦 = {𝑥 ∣ ¬ 𝑥𝑥}
43nex 1895 1 ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥𝑥}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wal 1650   = wceq 1652  wex 1874  {cab 2751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761
This theorem is referenced by:  bj-ru  33362
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