Theorem rru 3694

Description: Relative version of Russell's paradox ru 3696 (which corresponds to the case 𝐴 = V).

Originally a subproof in pwnss 5219. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid df-nel 3057. (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 variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq12 2842	. . . . . . 7 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
2	1	anidms 571	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
3	2	notbid 322	. . . . 5 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
4	3	cbvrabv 3405	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} = {𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝑦}
5	4	eleq2i 2844	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ↔ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝑦})
6		elex 3429	. . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝑦} → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ V)
7		elex 3429	. . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ V)
8	7	adantr 485	. . . 4 ⊢ (({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴 ∧ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥}) → {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ V)
9		eleq1 2840	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
10		id 22	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
11	10, 10	eleq12d 2847	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧))
12	11	notbid 322	. . . . . 6 ⊢ (𝑦 = 𝑧 → (¬ 𝑦 ∈ 𝑦 ↔ ¬ 𝑧 ∈ 𝑧))
13	9, 12	anbi12d 634	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑦) ↔ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑧)))
14		eleq1 2840	. . . . . 6 ⊢ (𝑧 = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} → (𝑧 ∈ 𝐴 ↔ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴))
15		eleq12 2842	. . . . . . . 8 ⊢ ((𝑧 = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∧ 𝑧 = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥}) → (𝑧 ∈ 𝑧 ↔ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥}))
16	15	anidms 571	. . . . . . 7 ⊢ (𝑧 = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} → (𝑧 ∈ 𝑧 ↔ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥}))
17	16	notbid 322	. . . . . 6 ⊢ (𝑧 = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} → (¬ 𝑧 ∈ 𝑧 ↔ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥}))
18	14, 17	anbi12d 634	. . . . 5 ⊢ (𝑧 = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} → ((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑧) ↔ ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴 ∧ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥})))
19		df-rab 3080	. . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝑦} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝑦)}
20	13, 18, 19	elab2gw 3587	. . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ V → ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝑦} ↔ ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴 ∧ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥})))
21	6, 8, 20	pm5.21nii 384	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝑦} ↔ ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴 ∧ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥}))
22	5, 21	bitri 278	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ↔ ({𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴 ∧ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥}))
23		pclem6 1024	. 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)
24	22, 23	ax-mp 5	1 ⊢ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  {crab 3075  Vcvv 3410

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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-rab 3080  df-v 3412

This theorem is referenced by:  pwnss  5219