Metamath Proof Explorer


Theorem rru

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

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

Ref Expression
Assertion rru ¬ { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∈ 𝐴

Proof

Step Hyp Ref Expression
1 eleq12 ( ( 𝑦 = { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∧ 𝑦 = { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ) → ( 𝑦𝑦 ↔ { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∈ { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ) )
2 1 anidms ( 𝑦 = { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } → ( 𝑦𝑦 ↔ { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∈ { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ) )
3 2 notbid ( 𝑦 = { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } → ( ¬ 𝑦𝑦 ↔ ¬ { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∈ { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ) )
4 eleq12 ( ( 𝑥 = 𝑦𝑥 = 𝑦 ) → ( 𝑥𝑥𝑦𝑦 ) )
5 4 anidms ( 𝑥 = 𝑦 → ( 𝑥𝑥𝑦𝑦 ) )
6 5 notbid ( 𝑥 = 𝑦 → ( ¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦 ) )
7 6 cbvrabv { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } = { 𝑦𝐴 ∣ ¬ 𝑦𝑦 }
8 3 7 elrab2 ( { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∈ { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ↔ ( { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∈ 𝐴 ∧ ¬ { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∈ { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ) )
9 pclem6 ( ( { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∈ { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ↔ ( { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∈ 𝐴 ∧ ¬ { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∈ { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ) ) → ¬ { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∈ 𝐴 )
10 8 9 ax-mp ¬ { 𝑥𝐴 ∣ ¬ 𝑥𝑥 } ∈ 𝐴