Metamath Proof Explorer


Theorem rusbcALT

Description: A version of Russell's paradox which is proven using proper substitution. (Contributed by Andrew Salmon, 18-Jun-2011) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Assertion rusbcALT x | x x V

Proof

Step Hyp Ref Expression
1 pm5.19 ¬ x | x x x | x x ¬ x | x x x | x x
2 sbcnel12g x | x x V [˙ x | x x / x]˙ x x x | x x / x x x | x x / x x
3 sbc8g x | x x V [˙ x | x x / x]˙ x x x | x x x | x x
4 df-nel x | x x / x x x | x x / x x ¬ x | x x / x x x | x x / x x
5 csbvarg x | x x V x | x x / x x = x | x x
6 5 5 eleq12d x | x x V x | x x / x x x | x x / x x x | x x x | x x
7 6 notbid x | x x V ¬ x | x x / x x x | x x / x x ¬ x | x x x | x x
8 4 7 syl5bb x | x x V x | x x / x x x | x x / x x ¬ x | x x x | x x
9 2 3 8 3bitr3d x | x x V x | x x x | x x ¬ x | x x x | x x
10 1 9 mto ¬ x | x x V
11 df-nel x | x x V ¬ x | x x V
12 10 11 mpbir x | x x V