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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 ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem rru
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq12 2842 . . . . . . 7 ((𝑥 = 𝑦𝑥 = 𝑦) → (𝑥𝑥𝑦𝑦))
21anidms 571 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
32notbid 322 . . . . 5 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
43cbvrabv 3405 . . . 4 {𝑥𝐴 ∣ ¬ 𝑥𝑥} = {𝑦𝐴 ∣ ¬ 𝑦𝑦}
54eleq2i 2844 . . 3 ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ↔ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑦𝐴 ∣ ¬ 𝑦𝑦})
6 elex 3429 . . . 4 ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑦𝐴 ∣ ¬ 𝑦𝑦} → {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ V)
7 elex 3429 . . . . 5 ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴 → {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ V)
87adantr 485 . . . 4 (({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴 ∧ ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥}) → {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ V)
9 eleq1 2840 . . . . . 6 (𝑦 = 𝑧 → (𝑦𝐴𝑧𝐴))
10 id 22 . . . . . . . 8 (𝑦 = 𝑧𝑦 = 𝑧)
1110, 10eleq12d 2847 . . . . . . 7 (𝑦 = 𝑧 → (𝑦𝑦𝑧𝑧))
1211notbid 322 . . . . . 6 (𝑦 = 𝑧 → (¬ 𝑦𝑦 ↔ ¬ 𝑧𝑧))
139, 12anbi12d 634 . . . . 5 (𝑦 = 𝑧 → ((𝑦𝐴 ∧ ¬ 𝑦𝑦) ↔ (𝑧𝐴 ∧ ¬ 𝑧𝑧)))
14 eleq1 2840 . . . . . 6 (𝑧 = {𝑥𝐴 ∣ ¬ 𝑥𝑥} → (𝑧𝐴 ↔ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴))
15 eleq12 2842 . . . . . . . 8 ((𝑧 = {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∧ 𝑧 = {𝑥𝐴 ∣ ¬ 𝑥𝑥}) → (𝑧𝑧 ↔ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥}))
1615anidms 571 . . . . . . 7 (𝑧 = {𝑥𝐴 ∣ ¬ 𝑥𝑥} → (𝑧𝑧 ↔ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥}))
1716notbid 322 . . . . . 6 (𝑧 = {𝑥𝐴 ∣ ¬ 𝑥𝑥} → (¬ 𝑧𝑧 ↔ ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥}))
1814, 17anbi12d 634 . . . . 5 (𝑧 = {𝑥𝐴 ∣ ¬ 𝑥𝑥} → ((𝑧𝐴 ∧ ¬ 𝑧𝑧) ↔ ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴 ∧ ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥})))
19 df-rab 3080 . . . . 5 {𝑦𝐴 ∣ ¬ 𝑦𝑦} = {𝑦 ∣ (𝑦𝐴 ∧ ¬ 𝑦𝑦)}
2013, 18, 19elab2gw 3587 . . . 4 ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ V → ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑦𝐴 ∣ ¬ 𝑦𝑦} ↔ ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴 ∧ ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥})))
216, 8, 20pm5.21nii 384 . . 3 ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑦𝐴 ∣ ¬ 𝑦𝑦} ↔ ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴 ∧ ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥}))
225, 21bitri 278 . 2 ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ↔ ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴 ∧ ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥}))
23 pclem6 1024 . 2 (({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ↔ ({𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴 ∧ ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ {𝑥𝐴 ∣ ¬ 𝑥𝑥})) → ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴)
2422, 23ax-mp 5 1 ¬ {𝑥𝐴 ∣ ¬ 𝑥𝑥} ∈ 𝐴
 Colors of variables: wff setvar class 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
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