Theorem bj-ru1 34818

Description: A version of Russell's paradox ru 3682 (see also bj-ru 34819). (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 34817	. . 3 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥)
2		abeq2 2862	. . 3 ⊢ (𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥))
3	1, 2	mtbir 326	. 2 ⊢ ¬ 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}
4	3	nex 1808	1 ⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}

Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1541   = wceq 1543  ∃wex 1787  {cab 2714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809

This theorem is referenced by:  bj-ru  34819