axioma de conjunts

Axioma:

[Ax][ x [&] {y_{1},...,x,...,y_{n}} = 0 ]

Teorema:

¬( x€x )

Demostració: ( Per absurd )

x€x

x€x & x€{y_{1},...,x,...y_{n}}

x€ x [&] {y_{1},...,x,...y_{n}}

x [&] {y_{1},...,x,...y_{n}} != 0

Teorema:

¬( x€y & y€x )

Demostració: ( Per absurd )

x€y & y€x

{x} [<< y & {y} [<< x

{x,y} [<< y [ || ] x

{x,y} = {x,y} [&] (y [ || ] x) = ({x,y} [&] y) [ || ] ({x,y} [&] x) = 0 [ || ] 0 = 0

{x} = 0 & {y} = 0

x€0 & y€0