Theorem bj-ru1 37607

Description: A version of Russell's paradox ru 3742 not mentioning the universal class. (see also bj-ru 37608). (Contributed by BJ, 12-Oct-2019.) Remove usage of ax-10 2175, ax-11 2191, ax-12 2212 by using eqabbw 2835 following BTernaryTau's similar revision of ru 3742. (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 2161	. . 3 ⊢ ¬ ∀𝑧(𝑧 ∈ 𝑦 ↔ ¬ 𝑧 ∈ 𝑧)
2		id 23	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
3	2, 2	eleq12d 2856	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑥 ↔ 𝑧 ∈ 𝑧))
4	3	notbid 321	. . . 4 ⊢ (𝑥 = 𝑧 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑧 ∈ 𝑧))
5	4	eqabbw 2835	. . 3 ⊢ (𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ ¬ 𝑧 ∈ 𝑧))
6	1, 5	mtbir 326	. 2 ⊢ ¬ 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}
7	6	nex 1829	1 ⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  ∀wal 1567   = wceq 1569  ∃wex 1808  {cab 2740

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837

This theorem is used by:  bj-ru  37608