Theorem bj-ru1 34254

Description: A version of Russell's paradox ru 3770 (see also bj-ru 34255). (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ru1	⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}

Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-ru1

Step	Hyp	Ref	Expression
1		bj-ru0 34253	. . 3 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥)
2		abeq2 2945	. . 3 ⊢ (𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥))
3	1, 2	mtbir 325	. 2 ⊢ ¬ 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}
4	3	nex 1797	1 ⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}

Syntax hints:  ¬ wn 3   ↔ wb 208  ∀wal 1531   = wceq 1533  ∃wex 1776  {cab 2799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893

This theorem is referenced by:  bj-ru  34255