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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 10232 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℝ | |
3 | 2re 11282 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ | |
4 | 2, 3 | pm3.2i 470 | . . . . . . . . . . 11 ⊢ (0 ∈ ℝ ∧ 2 ∈ ℝ) |
5 | 2 | leidi 10754 | . . . . . . . . . . . 12 ⊢ 0 ≤ 0 |
6 | 1le2 11433 | . . . . . . . . . . . 12 ⊢ 1 ≤ 2 | |
7 | 5, 6 | pm3.2i 470 | . . . . . . . . . . 11 ⊢ (0 ≤ 0 ∧ 1 ≤ 2) |
8 | iccss 12434 | . . . . . . . . . . 11 ⊢ (((0 ∈ ℝ ∧ 2 ∈ ℝ) ∧ (0 ≤ 0 ∧ 1 ≤ 2)) → (0[,]1) ⊆ (0[,]2)) | |
9 | 4, 7, 8 | mp2an 710 | . . . . . . . . . 10 ⊢ (0[,]1) ⊆ (0[,]2) |
10 | id 22 | . . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ (0[,]1)) | |
11 | 9, 10 | sseldi 3742 | . . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ (0[,]2)) |
12 | salgensscntex.a | . . . . . . . . 9 ⊢ 𝐴 = (0[,]2) | |
13 | 11, 12 | syl6eleqr 2850 | . . . . . . . 8 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ 𝐴) |
14 | snelpwi 5061 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ∈ 𝒫 𝐴) | |
15 | 13, 14 | syl 17 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]1) → {𝑦} ∈ 𝒫 𝐴) |
16 | snfi 8203 | . . . . . . . . . 10 ⊢ {𝑦} ∈ Fin | |
17 | fict 8723 | . . . . . . . . . 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 555 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]1) → ({𝑦} ∈ 𝒫 𝐴 ∧ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
23 | breq1 4807 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → (𝑥 ≼ ω ↔ {𝑦} ≼ ω)) | |
24 | difeq2 3865 | . . . . . . . . 9 ⊢ (𝑥 = {𝑦} → (𝐴 ∖ 𝑥) = (𝐴 ∖ {𝑦})) | |
25 | 24 | breq1d 4814 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ {𝑦}) ≼ ω)) |
26 | 23, 25 | orbi12d 748 | . . . . . . 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 3507 | . . . . . 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 3060 | . . . 4 ⊢ ∀𝑦 ∈ (0[,]1){𝑦} ∈ 𝑆 |
31 | eqid 2760 | . . . . 5 ⊢ (𝑦 ∈ (0[,]1) ↦ {𝑦}) = (𝑦 ∈ (0[,]1) ↦ {𝑦}) | |
32 | 31 | rnmptss 6555 | . . . 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 3776 | . 2 ⊢ 𝑋 ⊆ 𝑆 |
35 | ovex 6841 | . . . . . 6 ⊢ (0[,]2) ∈ V | |
36 | 12, 35 | eqeltri 2835 | . . . . 5 ⊢ 𝐴 ∈ V |
37 | 36 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐴 ∈ V) |
38 | 37, 27 | salexct 41055 | . . 3 ⊢ (⊤ → 𝑆 ∈ SAlg) |
39 | 38 | trud 1642 | . 2 ⊢ 𝑆 ∈ SAlg |
40 | ovex 6841 | . . . . . . . . 9 ⊢ (0[,]1) ∈ V | |
41 | 40 | mptex 6650 | . . . . . . . 8 ⊢ (𝑦 ∈ (0[,]1) ↦ {𝑦}) ∈ V |
42 | 41 | rnex 7265 | . . . . . . 7 ⊢ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) ∈ V |
43 | 1, 42 | eqeltri 2835 | . . . . . 6 ⊢ 𝑋 ∈ V |
44 | 43 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝑋 ∈ V) |
45 | salgensscntex.g | . . . . 5 ⊢ 𝐺 = (SalGen‘𝑋) | |
46 | 1 | unieqi 4597 | . . . . . 6 ⊢ ∪ 𝑋 = ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) |
47 | snex 5057 | . . . . . . . . 9 ⊢ {𝑦} ∈ V | |
48 | 47 | rgenw 3062 | . . . . . . . 8 ⊢ ∀𝑦 ∈ (0[,]1){𝑦} ∈ V |
49 | dfiun3g 5533 | . . . . . . . 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 2769 | . . . . . 6 ⊢ ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) = ∪ 𝑦 ∈ (0[,]1){𝑦} |
52 | iunid 4727 | . . . . . 6 ⊢ ∪ 𝑦 ∈ (0[,]1){𝑦} = (0[,]1) | |
53 | 46, 51, 52 | 3eqtrri 2787 | . . . . 5 ⊢ (0[,]1) = ∪ 𝑋 |
54 | 44, 45, 53 | unisalgen 41061 | . . . 4 ⊢ (⊤ → (0[,]1) ∈ 𝐺) |
55 | 54 | trud 1642 | . . 3 ⊢ (0[,]1) ∈ 𝐺 |
56 | eqid 2760 | . . . 4 ⊢ (0[,]1) = (0[,]1) | |
57 | 12, 27, 56 | salexct2 41060 | . . 3 ⊢ ¬ (0[,]1) ∈ 𝑆 |
58 | nelss 3805 | . . 3 ⊢ (((0[,]1) ∈ 𝐺 ∧ ¬ (0[,]1) ∈ 𝑆) → ¬ 𝐺 ⊆ 𝑆) | |
59 | 55, 57, 58 | mp2an 710 | . 2 ⊢ ¬ 𝐺 ⊆ 𝑆 |
60 | 34, 39, 59 | 3pm3.2i 1424 | 1 ⊢ (𝑋 ⊆ 𝑆 ∧ 𝑆 ∈ SAlg ∧ ¬ 𝐺 ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 382 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ⊤wtru 1633 ∈ wcel 2139 ∀wral 3050 {crab 3054 Vcvv 3340 ∖ cdif 3712 ⊆ wss 3715 𝒫 cpw 4302 {csn 4321 ∪ cuni 4588 ∪ ciun 4672 class class class wbr 4804 ↦ cmpt 4881 ran crn 5267 ‘cfv 6049 (class class class)co 6813 ωcom 7230 ≼ cdom 8119 Fincfn 8121 ℝcr 10127 0cc0 10128 1c1 10129 ≤ cle 10267 2c2 11262 [,]cicc 12371 SAlgcsalg 41031 SalGencsalgen 41035 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 ax-cc 9449 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-omul 7734 df-er 7911 df-map 8025 df-pm 8026 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-sup 8513 df-inf 8514 df-oi 8580 df-card 8955 df-acn 8958 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-n0 11485 df-z 11570 df-uz 11880 df-q 11982 df-rp 12026 df-xneg 12139 df-xadd 12140 df-xmul 12141 df-ioo 12372 df-ioc 12373 df-ico 12374 df-icc 12375 df-fz 12520 df-fzo 12660 df-fl 12787 df-seq 12996 df-exp 13055 df-hash 13312 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-limsup 14401 df-clim 14418 df-rlim 14419 df-sum 14616 df-topgen 16306 df-psmet 19940 df-xmet 19941 df-met 19942 df-bl 19943 df-mopn 19944 df-top 20901 df-topon 20918 df-bases 20952 df-ntr 21026 df-salg 41032 df-salgen 41036 |
This theorem is referenced by: (None) |
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