Theorem bj-ru1 36966

Description: A version of Russell's paradox ru 3768 not mentioning the universal class. (see also bj-ru 36967). (Contributed by BJ, 12-Oct-2019.) Remove usage of ax-10 2142, ax-11 2158, ax-12 2178 by using eqabbw 2809 following BTernaryTau's similar revision of ru 3768. (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 2128	. . 3 ⊢ ¬ ∀𝑧(𝑧 ∈ 𝑦 ↔ ¬ 𝑧 ∈ 𝑧)
2		id 22	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
3	2, 2	eleq12d 2829	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑥 ↔ 𝑧 ∈ 𝑧))
4	3	notbid 318	. . . 4 ⊢ (𝑥 = 𝑧 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑧 ∈ 𝑧))
5	4	eqabbw 2809	. . 3 ⊢ (𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ ¬ 𝑧 ∈ 𝑧))
6	1, 5	mtbir 323	. 2 ⊢ ¬ 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}
7	6	nex 1800	1 ⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1538   = wceq 1540  ∃wex 1779  {cab 2714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810

This theorem is referenced by:  bj-ru  36967