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Theorem fin23 8060
Description: Every II-finite set (every chain of subsets has a maximal element) is III-finite (has no denumerable collection of subsets). The proof here is the only one I could find, from http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm619.pdf p.94 (writeup by Tarski, credited to Kuratowski). Translated into English and modern notation, the proof proceeds as follows (variables renamed for uniqueness):

Suppose for a contradiction that  A is a set which is II-finite but not III-finite.

For any countable sequence of distinct subsets  T of  A, we can form a decreasing sequence of non-empty subsets  ( U `  T ) by taking finite intersections of initial segments of  T while skipping over any element of  T which would cause the intersection to be empty.

By II-finiteness (as fin2i2 7989) this sequence contains its intersection, call it  Y; since by induction every subset in the sequence  U is non-empty, the intersection must be non-empty.

Suppose that an element  X of  T has non-empty intersection with  Y. Thus, said element has a non-empty intersection with the corresponding element of  U, therefore it was used in the construction of  U and all further elements of  U are subsets of  X, thus  X contains the  Y. That is, all elements of  X either contain  Y or are disjoint from it.

Since there are only two cases, there must exist an infinite subset of  T which uniformly either contain  Y or are disjoint from it. In the former case we can create an infinite set by subtracting  Y from each element. In either case, call the result  Z; this is an infinite set of subsets of 
A, each of which is disjoint from  Y and contained in the union of  T; the union of 
Z is strictly contained in the union of  T, because only the latter is a superset of the non-empty set  Y.

The preceeding four steps may be iterated a countable number of times starting from the assumed denumerable set of subsets to produce a denumerable sequence  B of the  T sets from each stage. Great caution is required to avoid ax-dc 8117 here; in particular an effective version of the pigeonhole principle (for aleph-null pigeons and 2 holes) is required. Since a denumerable set of subsets is assumed to exist, we can conclude  om  e.  _V without the axiom.

This  B sequence is strictly decreasing, thus it has no minimum, contradicting the first assumption. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)

Assertion
Ref Expression
fin23  |-  ( A  e. FinII  ->  A  e. FinIII )

Proof of Theorem fin23
Dummy variables  a 
g  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isf33lem 8037 . 2  |- FinIII  =  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) }
21fin23lem40 8022 1  |-  ( A  e. FinII  ->  A  e. FinIII )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1701  FinIIcfin2 7950  FinIIIcfin3 7952
This theorem is referenced by:  fin1a2s  8085  finngch  8322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-rpss 6319  df-riota 6346  df-recs 6430  df-rdg 6465  df-seqom 6502  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-wdom 7318  df-card 7617  df-fin2 7957  df-fin4 7958  df-fin3 7959
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