(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Hi!

I want to show that lXl<lYl implies lXl[itex]\in[/itex]lYl where lXl and lYl are some cardinal numbers of two sets X and Y and the ordering < is defined on cardinal numbers .

2. Relevant equations

3. The attempt at a solution

I tried to solve it by myself as follows:

lXl < lYl [itex]\rightarrow[/itex] lXl[itex]\leq[/itex]lYl and not lXl=lYl( X is not equipotent to Y)

[itex]\rightarrow[/itex] there is a function f on X into Y s.t. f is a 1-1 function, and

not lXl=lYl( cardinal numbers lXl and lYl are not same)

[itex]\rightarrow[/itex] there is a function f on X into Y s.t. f is a 1-1 function, and

lXl[itex]\in[/itex]lYl or lYl[itex]\in[/itex]lXl since lXl and

lYl are initial ordinals.

But I can't determine why lXl must belong to lYl.

Could you give me a hint??

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# Cardinality of set

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