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 ¬ x A | ¬ x x A

Proof

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