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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > salexct3 | Structured version Visualization version GIF version |
Description: An example of a sigma-algebra that's not closed under uncountable union. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
salexct3.a | ⊢ 𝐴 = (0[,]2) |
salexct3.s | ⊢ 𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω)} |
salexct3.x | ⊢ 𝑋 = ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) |
Ref | Expression |
---|---|
salexct3 | ⊢ (𝑆 ∈ SAlg ∧ 𝑋 ⊆ 𝑆 ∧ ¬ ∪ 𝑋 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | salexct3.a | . . . . . 6 ⊢ 𝐴 = (0[,]2) | |
2 | ovex 7435 | . . . . . 6 ⊢ (0[,]2) ∈ V | |
3 | 1, 2 | eqeltri 2821 | . . . . 5 ⊢ 𝐴 ∈ V |
4 | 3 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐴 ∈ V) |
5 | salexct3.s | . . . 4 ⊢ 𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω)} | |
6 | 4, 5 | salexct 45560 | . . 3 ⊢ (⊤ → 𝑆 ∈ SAlg) |
7 | 6 | mptru 1540 | . 2 ⊢ 𝑆 ∈ SAlg |
8 | salexct3.x | . . 3 ⊢ 𝑋 = ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) | |
9 | 0re 11214 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℝ | |
10 | 2re 12284 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ | |
11 | 9, 10 | pm3.2i 470 | . . . . . . . . . . 11 ⊢ (0 ∈ ℝ ∧ 2 ∈ ℝ) |
12 | 9 | leidi 11746 | . . . . . . . . . . . 12 ⊢ 0 ≤ 0 |
13 | 1le2 12419 | . . . . . . . . . . . 12 ⊢ 1 ≤ 2 | |
14 | 12, 13 | pm3.2i 470 | . . . . . . . . . . 11 ⊢ (0 ≤ 0 ∧ 1 ≤ 2) |
15 | iccss 13390 | . . . . . . . . . . 11 ⊢ (((0 ∈ ℝ ∧ 2 ∈ ℝ) ∧ (0 ≤ 0 ∧ 1 ≤ 2)) → (0[,]1) ⊆ (0[,]2)) | |
16 | 11, 14, 15 | mp2an 689 | . . . . . . . . . 10 ⊢ (0[,]1) ⊆ (0[,]2) |
17 | id 22 | . . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ (0[,]1)) | |
18 | 16, 17 | sselid 3973 | . . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ (0[,]2)) |
19 | 18, 1 | eleqtrrdi 2836 | . . . . . . . 8 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ 𝐴) |
20 | snelpwi 5434 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ∈ 𝒫 𝐴) | |
21 | 19, 20 | syl 17 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]1) → {𝑦} ∈ 𝒫 𝐴) |
22 | snfi 9041 | . . . . . . . . . 10 ⊢ {𝑦} ∈ Fin | |
23 | fict 9645 | . . . . . . . . . 10 ⊢ ({𝑦} ∈ Fin → {𝑦} ≼ ω) | |
24 | 22, 23 | ax-mp 5 | . . . . . . . . 9 ⊢ {𝑦} ≼ ω |
25 | orc 864 | . . . . . . . . 9 ⊢ ({𝑦} ≼ ω → ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω)) | |
26 | 24, 25 | ax-mp 5 | . . . . . . . 8 ⊢ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω) |
27 | 26 | a1i 11 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]1) → ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω)) |
28 | 21, 27 | jca 511 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]1) → ({𝑦} ∈ 𝒫 𝐴 ∧ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
29 | breq1 5142 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → (𝑥 ≼ ω ↔ {𝑦} ≼ ω)) | |
30 | difeq2 4109 | . . . . . . . . 9 ⊢ (𝑥 = {𝑦} → (𝐴 ∖ 𝑥) = (𝐴 ∖ {𝑦})) | |
31 | 30 | breq1d 5149 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ {𝑦}) ≼ ω)) |
32 | 29, 31 | orbi12d 915 | . . . . . . 7 ⊢ (𝑥 = {𝑦} → ((𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω) ↔ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
33 | 32, 5 | elrab2 3679 | . . . . . 6 ⊢ ({𝑦} ∈ 𝑆 ↔ ({𝑦} ∈ 𝒫 𝐴 ∧ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
34 | 28, 33 | sylibr 233 | . . . . 5 ⊢ (𝑦 ∈ (0[,]1) → {𝑦} ∈ 𝑆) |
35 | 34 | rgen 3055 | . . . 4 ⊢ ∀𝑦 ∈ (0[,]1){𝑦} ∈ 𝑆 |
36 | eqid 2724 | . . . . 5 ⊢ (𝑦 ∈ (0[,]1) ↦ {𝑦}) = (𝑦 ∈ (0[,]1) ↦ {𝑦}) | |
37 | 36 | rnmptss 7115 | . . . 4 ⊢ (∀𝑦 ∈ (0[,]1){𝑦} ∈ 𝑆 → ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) ⊆ 𝑆) |
38 | 35, 37 | ax-mp 5 | . . 3 ⊢ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) ⊆ 𝑆 |
39 | 8, 38 | eqsstri 4009 | . 2 ⊢ 𝑋 ⊆ 𝑆 |
40 | 8 | unieqi 4912 | . . . . 5 ⊢ ∪ 𝑋 = ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) |
41 | vsnex 5420 | . . . . . . . 8 ⊢ {𝑦} ∈ V | |
42 | 41 | rgenw 3057 | . . . . . . 7 ⊢ ∀𝑦 ∈ (0[,]1){𝑦} ∈ V |
43 | dfiun3g 5954 | . . . . . . 7 ⊢ (∀𝑦 ∈ (0[,]1){𝑦} ∈ V → ∪ 𝑦 ∈ (0[,]1){𝑦} = ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦})) | |
44 | 42, 43 | ax-mp 5 | . . . . . 6 ⊢ ∪ 𝑦 ∈ (0[,]1){𝑦} = ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) |
45 | 44 | eqcomi 2733 | . . . . 5 ⊢ ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) = ∪ 𝑦 ∈ (0[,]1){𝑦} |
46 | iunid 5054 | . . . . 5 ⊢ ∪ 𝑦 ∈ (0[,]1){𝑦} = (0[,]1) | |
47 | 40, 45, 46 | 3eqtrri 2757 | . . . 4 ⊢ (0[,]1) = ∪ 𝑋 |
48 | 47 | eqcomi 2733 | . . 3 ⊢ ∪ 𝑋 = (0[,]1) |
49 | 1, 5, 48 | salexct2 45565 | . 2 ⊢ ¬ ∪ 𝑋 ∈ 𝑆 |
50 | 7, 39, 49 | 3pm3.2i 1336 | 1 ⊢ (𝑆 ∈ SAlg ∧ 𝑋 ⊆ 𝑆 ∧ ¬ ∪ 𝑋 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ∨ wo 844 ∧ w3a 1084 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ∀wral 3053 {crab 3424 Vcvv 3466 ∖ cdif 3938 ⊆ wss 3941 𝒫 cpw 4595 {csn 4621 ∪ cuni 4900 ∪ ciun 4988 class class class wbr 5139 ↦ cmpt 5222 ran crn 5668 (class class class)co 7402 ωcom 7849 ≼ cdom 8934 Fincfn 8936 ℝcr 11106 0cc0 11107 1c1 11108 ≤ cle 11247 2c2 12265 [,]cicc 13325 SAlgcsalg 45534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 ax-cc 10427 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-acn 9934 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-3 12274 df-n0 12471 df-z 12557 df-uz 12821 df-q 12931 df-rp 12973 df-xneg 13090 df-xadd 13091 df-xmul 13092 df-ioo 13326 df-ioc 13327 df-ico 13328 df-icc 13329 df-fz 13483 df-fzo 13626 df-fl 13755 df-seq 13965 df-exp 14026 df-hash 14289 df-cj 15044 df-re 15045 df-im 15046 df-sqrt 15180 df-abs 15181 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15631 df-topgen 17390 df-psmet 21222 df-xmet 21223 df-met 21224 df-bl 21225 df-mopn 21226 df-top 22720 df-topon 22737 df-bases 22773 df-ntr 22848 df-salg 45535 |
This theorem is referenced by: (None) |
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