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Theorem fin23 8570
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 nonempty 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 8499) this sequence contains its intersection, call it  Y; since by induction every subset in the sequence  U is nonempty, the intersection must be nonempty.

Suppose that an element  X of  T has nonempty intersection with  Y. Thus, said element has a nonempty 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 nonempty set  Y.

The preceding 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 8627 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 8547 . 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 8532 1  |-  ( A  e. FinII  ->  A  e. FinIII )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756  FinIIcfin2 8460  FinIIIcfin3 8462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-rpss 6372  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-seqom 6915  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-wdom 7786  df-card 8121  df-fin2 8467  df-fin4 8468  df-fin3 8469
This theorem is referenced by:  fin1a2s  8595  finngch  8834  fin2so  28428
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