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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 7439 | . 2 ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) = (𝐴(𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢)))(𝑈‘𝐴))) |
3 | fvex 6113 | . . 3 ⊢ (𝑈‘𝐴) ∈ V | |
4 | fveq2 6103 | . . . . . . 7 ⊢ (𝑖 = 𝐴 → (𝑡‘𝑖) = (𝑡‘𝐴)) | |
5 | 4 | ineq1d 3775 | . . . . . 6 ⊢ (𝑖 = 𝐴 → ((𝑡‘𝑖) ∩ 𝑢) = ((𝑡‘𝐴) ∩ 𝑢)) |
6 | 5 | eqeq1d 2612 | . . . . 5 ⊢ (𝑖 = 𝐴 → (((𝑡‘𝑖) ∩ 𝑢) = ∅ ↔ ((𝑡‘𝐴) ∩ 𝑢) = ∅)) |
7 | 6, 5 | ifbieq2d 4061 | . . . 4 ⊢ (𝑖 = 𝐴 → if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢)) = if(((𝑡‘𝐴) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝐴) ∩ 𝑢))) |
8 | ineq2 3770 | . . . . . 6 ⊢ (𝑢 = (𝑈‘𝐴) → ((𝑡‘𝐴) ∩ 𝑢) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴))) | |
9 | 8 | eqeq1d 2612 | . . . . 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 4063 | . . . 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 2610 | . . . 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 4728 | . . . . 5 ⊢ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) ∈ V |
14 | 3, 13 | ifex 4106 | . . . 4 ⊢ if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴))) ∈ V |
15 | 7, 11, 12, 14 | ovmpt2 6694 | . . 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 703 | . 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 2644 | 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 1475 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 ∅c0 3874 ifcif 4036 ∪ cuni 4372 ran crn 5039 suc csuc 5642 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ωcom 6957 seq𝜔cseqom 7429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seqom 7430 |
This theorem is referenced by: fin23lem13 9037 fin23lem14 9038 fin23lem19 9041 |
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