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Theorem rru 3704
Description: Relative version of Russell's paradox ru 3705 (which corresponds to the case 𝐴 = V).

Originally a subproof in pwnss 5141. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid df-nel 3091. (Revised by Steven Nguyen, 23-Nov-2022.)

Assertion
Ref Expression
rru ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem rru
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq12 2872 . . . . 5 ((𝑦 = {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∧ 𝑦 = {𝑥𝐴 ∣ ¬ 𝑥𝑥}) → (𝑦𝑦 ↔ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥}))
21anidms 567 . . . 4 (𝑦 = {𝑥𝐴 ∣ ¬ 𝑥𝑥} → (𝑦𝑦 ↔ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥}))
32notbid 319 . . 3 (𝑦 = {𝑥𝐴 ∣ ¬ 𝑥𝑥} → (¬ 𝑦𝑦 ↔ ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥}))
4 eleq12 2872 . . . . . 6 ((𝑥 = 𝑦𝑥 = 𝑦) → (𝑥𝑥𝑦𝑦))
54anidms 567 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
65notbid 319 . . . 4 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
76cbvrabv 3434 . . 3 {𝑥𝐴 ∣ ¬ 𝑥𝑥} = {𝑦𝐴 ∣ ¬ 𝑦𝑦}
83, 7elrab2 3621 . 2 ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ↔ ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴 ∧ ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥}))
9 pclem6 1020 . 2 (({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ↔ ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴 ∧ ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥})) → ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴)
108, 9ax-mp 5 1 ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396   = wceq 1522  wcel 2081  {crab 3109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-rab 3114  df-v 3439
This theorem is referenced by:  pwnss  5141
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