MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin23 Structured version   Unicode version

Theorem fin23 8765
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 8694) 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 8822 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 8742 . 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 8727 1  |-  ( A  e. FinII  ->  A  e. FinIII )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767  FinIIcfin2 8655  FinIIIcfin3 8657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-rpss 6562  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-seqom 7110  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-wdom 7981  df-card 8316  df-fin2 8662  df-fin4 8663  df-fin3 8664
This theorem is referenced by:  fin1a2s  8790  finngch  9029  fin2so  29614
  Copyright terms: Public domain W3C validator