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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > salgensscntex | Structured version Visualization version GIF version |
Description: This counterexample shows that the sigma-algebra generated by a set is not the smallest sigma-algebra containing the set, if we consider also sigma-algebras with a larger base set. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
salgensscntex.a | ⊢ 𝐴 = (0[,]2) |
salgensscntex.s | ⊢ 𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω)} |
salgensscntex.x | ⊢ 𝑋 = ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) |
salgensscntex.g | ⊢ 𝐺 = (SalGen‘𝑋) |
Ref | Expression |
---|---|
salgensscntex | ⊢ (𝑋 ⊆ 𝑆 ∧ 𝑆 ∈ SAlg ∧ ¬ 𝐺 ⊆ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | salgensscntex.x | . . 3 ⊢ 𝑋 = ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) | |
2 | 0re 10078 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℝ | |
3 | 2re 11128 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ | |
4 | 2, 3 | pm3.2i 470 | . . . . . . . . . . 11 ⊢ (0 ∈ ℝ ∧ 2 ∈ ℝ) |
5 | 2 | leidi 10600 | . . . . . . . . . . . 12 ⊢ 0 ≤ 0 |
6 | 1le2 11279 | . . . . . . . . . . . 12 ⊢ 1 ≤ 2 | |
7 | 5, 6 | pm3.2i 470 | . . . . . . . . . . 11 ⊢ (0 ≤ 0 ∧ 1 ≤ 2) |
8 | iccss 12279 | . . . . . . . . . . 11 ⊢ (((0 ∈ ℝ ∧ 2 ∈ ℝ) ∧ (0 ≤ 0 ∧ 1 ≤ 2)) → (0[,]1) ⊆ (0[,]2)) | |
9 | 4, 7, 8 | mp2an 708 | . . . . . . . . . 10 ⊢ (0[,]1) ⊆ (0[,]2) |
10 | id 22 | . . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ (0[,]1)) | |
11 | 9, 10 | sseldi 3634 | . . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ (0[,]2)) |
12 | salgensscntex.a | . . . . . . . . 9 ⊢ 𝐴 = (0[,]2) | |
13 | 11, 12 | syl6eleqr 2741 | . . . . . . . 8 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ 𝐴) |
14 | snelpwi 4942 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ∈ 𝒫 𝐴) | |
15 | 13, 14 | syl 17 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]1) → {𝑦} ∈ 𝒫 𝐴) |
16 | snfi 8079 | . . . . . . . . . 10 ⊢ {𝑦} ∈ Fin | |
17 | fict 8588 | . . . . . . . . . 10 ⊢ ({𝑦} ∈ Fin → {𝑦} ≼ ω) | |
18 | 16, 17 | ax-mp 5 | . . . . . . . . 9 ⊢ {𝑦} ≼ ω |
19 | orc 399 | . . . . . . . . 9 ⊢ ({𝑦} ≼ ω → ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω)) | |
20 | 18, 19 | ax-mp 5 | . . . . . . . 8 ⊢ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω) |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]1) → ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω)) |
22 | 15, 21 | jca 553 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]1) → ({𝑦} ∈ 𝒫 𝐴 ∧ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
23 | breq1 4688 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → (𝑥 ≼ ω ↔ {𝑦} ≼ ω)) | |
24 | difeq2 3755 | . . . . . . . . 9 ⊢ (𝑥 = {𝑦} → (𝐴 ∖ 𝑥) = (𝐴 ∖ {𝑦})) | |
25 | 24 | breq1d 4695 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ {𝑦}) ≼ ω)) |
26 | 23, 25 | orbi12d 746 | . . . . . . 7 ⊢ (𝑥 = {𝑦} → ((𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω) ↔ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
27 | salgensscntex.s | . . . . . . 7 ⊢ 𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω)} | |
28 | 26, 27 | elrab2 3399 | . . . . . 6 ⊢ ({𝑦} ∈ 𝑆 ↔ ({𝑦} ∈ 𝒫 𝐴 ∧ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
29 | 22, 28 | sylibr 224 | . . . . 5 ⊢ (𝑦 ∈ (0[,]1) → {𝑦} ∈ 𝑆) |
30 | 29 | rgen 2951 | . . . 4 ⊢ ∀𝑦 ∈ (0[,]1){𝑦} ∈ 𝑆 |
31 | eqid 2651 | . . . . 5 ⊢ (𝑦 ∈ (0[,]1) ↦ {𝑦}) = (𝑦 ∈ (0[,]1) ↦ {𝑦}) | |
32 | 31 | rnmptss 6432 | . . . 4 ⊢ (∀𝑦 ∈ (0[,]1){𝑦} ∈ 𝑆 → ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) ⊆ 𝑆) |
33 | 30, 32 | ax-mp 5 | . . 3 ⊢ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) ⊆ 𝑆 |
34 | 1, 33 | eqsstri 3668 | . 2 ⊢ 𝑋 ⊆ 𝑆 |
35 | ovex 6718 | . . . . . 6 ⊢ (0[,]2) ∈ V | |
36 | 12, 35 | eqeltri 2726 | . . . . 5 ⊢ 𝐴 ∈ V |
37 | 36 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐴 ∈ V) |
38 | 37, 27 | salexct 40870 | . . 3 ⊢ (⊤ → 𝑆 ∈ SAlg) |
39 | 38 | trud 1533 | . 2 ⊢ 𝑆 ∈ SAlg |
40 | ovex 6718 | . . . . . . . . 9 ⊢ (0[,]1) ∈ V | |
41 | 40 | mptex 6527 | . . . . . . . 8 ⊢ (𝑦 ∈ (0[,]1) ↦ {𝑦}) ∈ V |
42 | 41 | rnex 7142 | . . . . . . 7 ⊢ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) ∈ V |
43 | 1, 42 | eqeltri 2726 | . . . . . 6 ⊢ 𝑋 ∈ V |
44 | 43 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝑋 ∈ V) |
45 | salgensscntex.g | . . . . 5 ⊢ 𝐺 = (SalGen‘𝑋) | |
46 | 1 | unieqi 4477 | . . . . . 6 ⊢ ∪ 𝑋 = ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) |
47 | snex 4938 | . . . . . . . . 9 ⊢ {𝑦} ∈ V | |
48 | 47 | rgenw 2953 | . . . . . . . 8 ⊢ ∀𝑦 ∈ (0[,]1){𝑦} ∈ V |
49 | dfiun3g 5410 | . . . . . . . 8 ⊢ (∀𝑦 ∈ (0[,]1){𝑦} ∈ V → ∪ 𝑦 ∈ (0[,]1){𝑦} = ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦})) | |
50 | 48, 49 | ax-mp 5 | . . . . . . 7 ⊢ ∪ 𝑦 ∈ (0[,]1){𝑦} = ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) |
51 | 50 | eqcomi 2660 | . . . . . 6 ⊢ ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) = ∪ 𝑦 ∈ (0[,]1){𝑦} |
52 | iunid 4607 | . . . . . 6 ⊢ ∪ 𝑦 ∈ (0[,]1){𝑦} = (0[,]1) | |
53 | 46, 51, 52 | 3eqtrri 2678 | . . . . 5 ⊢ (0[,]1) = ∪ 𝑋 |
54 | 44, 45, 53 | unisalgen 40876 | . . . 4 ⊢ (⊤ → (0[,]1) ∈ 𝐺) |
55 | 54 | trud 1533 | . . 3 ⊢ (0[,]1) ∈ 𝐺 |
56 | eqid 2651 | . . . 4 ⊢ (0[,]1) = (0[,]1) | |
57 | 12, 27, 56 | salexct2 40875 | . . 3 ⊢ ¬ (0[,]1) ∈ 𝑆 |
58 | nelss 3697 | . . 3 ⊢ (((0[,]1) ∈ 𝐺 ∧ ¬ (0[,]1) ∈ 𝑆) → ¬ 𝐺 ⊆ 𝑆) | |
59 | 55, 57, 58 | mp2an 708 | . 2 ⊢ ¬ 𝐺 ⊆ 𝑆 |
60 | 34, 39, 59 | 3pm3.2i 1259 | 1 ⊢ (𝑋 ⊆ 𝑆 ∧ 𝑆 ∈ SAlg ∧ ¬ 𝐺 ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 382 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ⊤wtru 1524 ∈ wcel 2030 ∀wral 2941 {crab 2945 Vcvv 3231 ∖ cdif 3604 ⊆ wss 3607 𝒫 cpw 4191 {csn 4210 ∪ cuni 4468 ∪ ciun 4552 class class class wbr 4685 ↦ cmpt 4762 ran crn 5144 ‘cfv 5926 (class class class)co 6690 ωcom 7107 ≼ cdom 7995 Fincfn 7997 ℝcr 9973 0cc0 9974 1c1 9975 ≤ cle 10113 2c2 11108 [,]cicc 12216 SAlgcsalg 40846 SalGencsalgen 40850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cc 9295 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-omul 7610 df-er 7787 df-map 7901 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-acn 8806 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-z 11416 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-ioc 12218 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-limsup 14246 df-clim 14263 df-rlim 14264 df-sum 14461 df-topgen 16151 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-top 20747 df-topon 20764 df-bases 20798 df-ntr 20872 df-salg 40847 df-salgen 40851 |
This theorem is referenced by: (None) |
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