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Mirrors > Home > MPE Home > Th. List > fin23 | Structured version Visualization version GIF version |
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 10250) 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 10378 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.) |
Ref | Expression |
---|---|
fin23 | ⊢ (𝐴 ∈ FinII → 𝐴 ∈ FinIII) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isf33lem 10298 | . 2 ⊢ FinIII = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} | |
2 | 1 | fin23lem40 10283 | 1 ⊢ (𝐴 ∈ FinII → 𝐴 ∈ FinIII) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 FinIIcfin2 10211 FinIIIcfin3 10213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-rpss 7656 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-seqom 8390 df-1o 8408 df-er 8644 df-map 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-wdom 9497 df-card 9871 df-fin2 10218 df-fin4 10219 df-fin3 10220 |
This theorem is referenced by: fin1a2s 10346 finngch 10587 fin2so 36032 |
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