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 | ⊢ ¬ { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ∈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq12 | ⊢ ( ( 𝑦 = { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ∧ 𝑦 = { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ) → ( 𝑦 ∈ 𝑦 ↔ { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ∈ { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ) ) | |
2 | 1 | anidms | ⊢ ( 𝑦 = { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } → ( 𝑦 ∈ 𝑦 ↔ { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ∈ { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ) ) |
3 | 2 | notbid | ⊢ ( 𝑦 = { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } → ( ¬ 𝑦 ∈ 𝑦 ↔ ¬ { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ∈ { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ) ) |
4 | eleq12 | ⊢ ( ( 𝑥 = 𝑦 ∧ 𝑥 = 𝑦 ) → ( 𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦 ) ) | |
5 | 4 | anidms | ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦 ) ) |
6 | 5 | notbid | ⊢ ( 𝑥 = 𝑦 → ( ¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦 ) ) |
7 | 6 | cbvrabv | ⊢ { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } = { 𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝑦 } |
8 | 3 7 | elrab2 | ⊢ ( { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ∈ { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ↔ ( { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ∈ 𝐴 ∧ ¬ { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ∈ { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ) ) |
9 | pclem6 | ⊢ ( ( { 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 | ⊢ ¬ { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥 } ∈ 𝐴 |