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Mirrors > Home > MPE Home > Th. List > fin23lem12 | Structured version Visualization version GIF version |
Description: The beginning of the
proof that every II-finite set (every chain of
subsets has a maximal element) is III-finite (has no denumerable
collection of subsets).
This first section is dedicated to the construction of 𝑈 and its intersection. First, the value of 𝑈 at a successor. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
Ref | Expression |
---|---|
fin23lem.a | ⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) |
Ref | Expression |
---|---|
fin23lem12 | ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fin23lem.a | . . 3 ⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) | |
2 | 1 | seqomsuc 8495 | . 2 ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) = (𝐴(𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢)))(𝑈‘𝐴))) |
3 | fvex 6919 | . . 3 ⊢ (𝑈‘𝐴) ∈ V | |
4 | fveq2 6906 | . . . . . . 7 ⊢ (𝑖 = 𝐴 → (𝑡‘𝑖) = (𝑡‘𝐴)) | |
5 | 4 | ineq1d 4226 | . . . . . 6 ⊢ (𝑖 = 𝐴 → ((𝑡‘𝑖) ∩ 𝑢) = ((𝑡‘𝐴) ∩ 𝑢)) |
6 | 5 | eqeq1d 2736 | . . . . 5 ⊢ (𝑖 = 𝐴 → (((𝑡‘𝑖) ∩ 𝑢) = ∅ ↔ ((𝑡‘𝐴) ∩ 𝑢) = ∅)) |
7 | 6, 5 | ifbieq2d 4556 | . . . 4 ⊢ (𝑖 = 𝐴 → if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢)) = if(((𝑡‘𝐴) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝐴) ∩ 𝑢))) |
8 | ineq2 4221 | . . . . . 6 ⊢ (𝑢 = (𝑈‘𝐴) → ((𝑡‘𝐴) ∩ 𝑢) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴))) | |
9 | 8 | eqeq1d 2736 | . . . . 5 ⊢ (𝑢 = (𝑈‘𝐴) → (((𝑡‘𝐴) ∩ 𝑢) = ∅ ↔ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅)) |
10 | id 22 | . . . . 5 ⊢ (𝑢 = (𝑈‘𝐴) → 𝑢 = (𝑈‘𝐴)) | |
11 | 9, 10, 8 | ifbieq12d 4558 | . . . 4 ⊢ (𝑢 = (𝑈‘𝐴) → if(((𝑡‘𝐴) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝐴) ∩ 𝑢)) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴)))) |
12 | eqid 2734 | . . . 4 ⊢ (𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))) = (𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))) | |
13 | 3 | inex2 5323 | . . . . 5 ⊢ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) ∈ V |
14 | 3, 13 | ifex 4580 | . . . 4 ⊢ if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴))) ∈ V |
15 | 7, 11, 12, 14 | ovmpo 7592 | . . 3 ⊢ ((𝐴 ∈ ω ∧ (𝑈‘𝐴) ∈ V) → (𝐴(𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢)))(𝑈‘𝐴)) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴)))) |
16 | 3, 15 | mpan2 691 | . 2 ⊢ (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢)))(𝑈‘𝐴)) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴)))) |
17 | 2, 16 | eqtrd 2774 | 1 ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∩ cin 3961 ∅c0 4338 ifcif 4530 ∪ cuni 4911 ran crn 5689 suc csuc 6387 ‘cfv 6562 (class class class)co 7430 ∈ cmpo 7432 ωcom 7886 seqωcseqom 8485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-seqom 8486 |
This theorem is referenced by: fin23lem13 10369 fin23lem14 10370 fin23lem19 10373 |
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