Theorem fin23 10401

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 10330) 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 10458 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 ω ∈ 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 ∈ FinII → 𝐴 ∈ 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 10378	. 2 ⊢ FinIII = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
2	1	fin23lem40 10363	1 ⊢ (𝐴 ∈ FinII → 𝐴 ∈ FinIII)

Syntax hints:   → wi 4   ∈ wcel 2108  Fin^IIcfin2 10291  Fin^IIIcfin3 10293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-isom 6539  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-rpss 7715  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-seqom 8460  df-1o 8478  df-er 8717  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-wdom 9577  df-card 9951  df-fin2 10298  df-fin4 10299  df-fin3 10300

This theorem is referenced by:  fin1a2s  10426  finngch  10667  fin2so  37577