Theorem bj-ru1 37366

Description: A version of Russell's paradox ru 3733 not mentioning the universal class. (see also bj-ru 37367). (Contributed by BJ, 12-Oct-2019.) Remove usage of ax-10 2165, ax-11 2181, ax-12 2202 by using eqabbw 2825 following BTernaryTau's similar revision of ru 3733. (Revised by BJ, 28-Jun-2025.) (Proof modification is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-ru1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ru0 2151	. . 3 ⊢ ¬ ∀𝑧(𝑧 ∈ 𝑦 ↔ ¬ 𝑧 ∈ 𝑧)
2		id 22	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
3	2, 2	eleq12d 2846	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑥 ↔ 𝑧 ∈ 𝑧))
4	3	notbid 320	. . . 4 ⊢ (𝑥 = 𝑧 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑧 ∈ 𝑧))
5	4	eqabbw 2825	. . 3 ⊢ (𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ ¬ 𝑧 ∈ 𝑧))
6	1, 5	mtbir 325	. 2 ⊢ ¬ 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}
7	6	nex 1810	1 ⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}

Syntax hints:  ¬ wn 3   ↔ wb 208  ∀wal 1548   = wceq 1550  ∃wex 1789  {cab 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827

This theorem is referenced by:  bj-ru  37367