@ means less numerous than or equally numerous with

# means less numerous than

~ means equinumerous

/x/ means cardinal of x

e means element of

€ means subset of

¥ is the intersection symbol

Theorem: (x @ y)(y @ x) --> x ~ y

Proof:

x # y --> No bijection exists from x to y; ran(x) ¥ y is non-empty.

So x e T, where T is the class of sets each of which is bijective with x.

So x ~ /x/ e T and /x/ e On U 0. Likewise for y.

From this, we see that

x @ y equiv. to /x/ <= /y/

Now /x/ <= /y/ --> /x/ € of /y/

and likewise for /y/ <= /x/

Thus (x@y)(y@x) -> (/x/ € /y/)(/y/ € /x/) -> /x/ = /y/ -> x ~ y

Nothing original. Just showing that SBT can be kept short.

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