Theorem rru 3714

Description: Relative version of Russell's paradox ru 3715 (which corresponds to the case 𝐴 = V).

Originally a subproof in pwnss 5272. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid df-nel 3050. (Revised by Steven Nguyen, 23-Nov-2022.) Reduce axiom usage. (Revised by Gino Giotto, 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 2828	. . . . 5 ⊢ ((𝑦 = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∧ 𝑦 = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥}) → (𝑦 ∈ 𝑦 ↔ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥}))
2	1	anidms 567	. . . 4 ⊢ (𝑦 = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} → (𝑦 ∈ 𝑦 ↔ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥}))
3	2	notbid 318	. . 3 ⊢ (𝑦 = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} → (¬ 𝑦 ∈ 𝑦 ↔ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥}))
4		eleq12 2828	. . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
5	4	anidms 567	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
6	5	notbid 318	. . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
7	6	cbvrabv 3426	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} = {𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝑦}
8	3, 7	elrab2 3627	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ↔ ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴 ∧ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥}))
9		pclem6 1023	. 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

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {crab 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434

This theorem is referenced by:  pwnss  5272