Theorem fin23 9211

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 9140) 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 9268 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 9188	. 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 9173	1 |- ( A e. FinII -> A e. FinIII )

Syntax hints:   -> wi 4   e. wcel 1990  Fin^IIcfin2 9101  Fin^IIIcfin3 9103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rpss 6937  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-wdom 8464  df-card 8765  df-fin2 9108  df-fin4 9109  df-fin3 9110

This theorem is referenced by:  fin1a2s  9236  finngch  9477  fin2so  33396