sorry, I also have this necessary assumption:

     5.  BIGUNION (IMAGE f 𝕌(:num)) = sp

> Il giorno 18 set 2018, alle ore 01:00, Chun Tian (binghe) 
> <binghe.l...@gmail.com> ha scritto:
> 
> Hi,
> 
> I ran into the following goal in the proof of a big theorem:
> 
>   ∃x. a ⊆ f x
>   ------------------------------------
>     3.  f 0 = ∅
>     4.  ∀n. f n ⊆ f (SUC n)
>    20.  ∀n. f n ⊆ sp
>    21.  a ⊆ sp
> 
> I have an increasing sequence of sets (f n) from empty to the whole space 
> (sp), and I have a single set “a” (⊆ sp). It looks *obvious* that, I can 
> always choose a big enough index x such that (f x) will contain “a”. But how 
> can I make this argument formal?
> 
> thanks,
> 
> Chun
> 

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