Theorem fin23 8837

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 8766) 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 8894 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.

Step	Hyp	Ref	Expression
1		isf33lem 8814	. 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 ) }
2	1	fin23lem40 8799	1 |- ( A e. FinII -> A e. FinIII )

Syntax hints:   -> wi 4   e. wcel 1904  Fin^IIcfin2 8727  Fin^IIIcfin3 8729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-rpss 6590  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-seqom 7183  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-wdom 8092  df-card 8391  df-fin2 8734  df-fin4 8735  df-fin3 8736

This theorem is referenced by:  fin1a2s  8862  finngch  9098  fin2so  31996