Theorem rru 3740

Description: Relative version of Russell's paradox ru 3741 (which corresponds to the case 𝐴 = V).

Originally a subproof in pwnss 5317. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid df-nel 3066. (Revised by Steven Nguyen, 23-Nov-2022.) Reduce axiom usage. (Revised by GG, 30-Aug-2024.)

Assertion

Ref	Expression
rru	⊢ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem rru

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq12 2854	. . . . 5 ⊢ ((𝑦 = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∧ 𝑦 = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥}) → (𝑦 ∈ 𝑦 ↔ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥}))
2	1	anidms 577	. . . 4 ⊢ (𝑦 = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} → (𝑦 ∈ 𝑦 ↔ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥}))
3	2	notbid 321	. . 3 ⊢ (𝑦 = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} → (¬ 𝑦 ∈ 𝑦 ↔ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥}))
4		eleq12 2854	. . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
5	4	anidms 577	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
6	5	notbid 321	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
7	6	cbvrabv 3426	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} = {𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝑦}
8	3, 7	elrab2 3652	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ↔ ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴 ∧ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥}))
9		pclem6 1043	. 2 ⊢ (({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ↔ ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴 ∧ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥})) → ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴)
10	8, 9	ax-mp 5	1 ⊢ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {crab 3416

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456

This theorem is used by:  pwnss  5317