![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 7168 | . . . . . 6 ⊢ (0[,]2) ∈ V | |
3 | 1, 2 | eqeltri 2886 | . . . . 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 42974 | . . 3 ⊢ (⊤ → 𝑆 ∈ SAlg) |
7 | 6 | mptru 1545 | . 2 ⊢ 𝑆 ∈ SAlg |
8 | salexct3.x | . . 3 ⊢ 𝑋 = ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) | |
9 | 0re 10632 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℝ | |
10 | 2re 11699 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ | |
11 | 9, 10 | pm3.2i 474 | . . . . . . . . . . 11 ⊢ (0 ∈ ℝ ∧ 2 ∈ ℝ) |
12 | 9 | leidi 11163 | . . . . . . . . . . . 12 ⊢ 0 ≤ 0 |
13 | 1le2 11834 | . . . . . . . . . . . 12 ⊢ 1 ≤ 2 | |
14 | 12, 13 | pm3.2i 474 | . . . . . . . . . . 11 ⊢ (0 ≤ 0 ∧ 1 ≤ 2) |
15 | iccss 12793 | . . . . . . . . . . 11 ⊢ (((0 ∈ ℝ ∧ 2 ∈ ℝ) ∧ (0 ≤ 0 ∧ 1 ≤ 2)) → (0[,]1) ⊆ (0[,]2)) | |
16 | 11, 14, 15 | mp2an 691 | . . . . . . . . . 10 ⊢ (0[,]1) ⊆ (0[,]2) |
17 | id 22 | . . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ (0[,]1)) | |
18 | 16, 17 | sseldi 3913 | . . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ (0[,]2)) |
19 | 18, 1 | eleqtrrdi 2901 | . . . . . . . 8 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ 𝐴) |
20 | snelpwi 5302 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ∈ 𝒫 𝐴) | |
21 | 19, 20 | syl 17 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]1) → {𝑦} ∈ 𝒫 𝐴) |
22 | snfi 8577 | . . . . . . . . . 10 ⊢ {𝑦} ∈ Fin | |
23 | fict 9100 | . . . . . . . . . 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 515 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]1) → ({𝑦} ∈ 𝒫 𝐴 ∧ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
29 | breq1 5033 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → (𝑥 ≼ ω ↔ {𝑦} ≼ ω)) | |
30 | difeq2 4044 | . . . . . . . . 9 ⊢ (𝑥 = {𝑦} → (𝐴 ∖ 𝑥) = (𝐴 ∖ {𝑦})) | |
31 | 30 | breq1d 5040 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ {𝑦}) ≼ ω)) |
32 | 29, 31 | orbi12d 916 | . . . . . . 7 ⊢ (𝑥 = {𝑦} → ((𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω) ↔ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
33 | 32, 5 | elrab2 3631 | . . . . . 6 ⊢ ({𝑦} ∈ 𝑆 ↔ ({𝑦} ∈ 𝒫 𝐴 ∧ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
34 | 28, 33 | sylibr 237 | . . . . 5 ⊢ (𝑦 ∈ (0[,]1) → {𝑦} ∈ 𝑆) |
35 | 34 | rgen 3116 | . . . 4 ⊢ ∀𝑦 ∈ (0[,]1){𝑦} ∈ 𝑆 |
36 | eqid 2798 | . . . . 5 ⊢ (𝑦 ∈ (0[,]1) ↦ {𝑦}) = (𝑦 ∈ (0[,]1) ↦ {𝑦}) | |
37 | 36 | rnmptss 6863 | . . . 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 3949 | . 2 ⊢ 𝑋 ⊆ 𝑆 |
40 | 8 | unieqi 4813 | . . . . 5 ⊢ ∪ 𝑋 = ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) |
41 | snex 5297 | . . . . . . . 8 ⊢ {𝑦} ∈ V | |
42 | 41 | rgenw 3118 | . . . . . . 7 ⊢ ∀𝑦 ∈ (0[,]1){𝑦} ∈ V |
43 | dfiun3g 5800 | . . . . . . 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 2807 | . . . . 5 ⊢ ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) = ∪ 𝑦 ∈ (0[,]1){𝑦} |
46 | iunid 4947 | . . . . 5 ⊢ ∪ 𝑦 ∈ (0[,]1){𝑦} = (0[,]1) | |
47 | 40, 45, 46 | 3eqtrri 2826 | . . . 4 ⊢ (0[,]1) = ∪ 𝑋 |
48 | 47 | eqcomi 2807 | . . 3 ⊢ ∪ 𝑋 = (0[,]1) |
49 | 1, 5, 48 | salexct2 42979 | . 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 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ⊤wtru 1539 ∈ wcel 2111 ∀wral 3106 {crab 3110 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 𝒫 cpw 4497 {csn 4525 ∪ cuni 4800 ∪ ciun 4881 class class class wbr 5030 ↦ cmpt 5110 ran crn 5520 (class class class)co 7135 ωcom 7560 ≼ cdom 8490 Fincfn 8492 ℝcr 10525 0cc0 10526 1c1 10527 ≤ cle 10665 2c2 11680 [,]cicc 12729 SAlgcsalg 42950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-omul 8090 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-acn 9355 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-top 21499 df-topon 21516 df-bases 21551 df-ntr 21625 df-salg 42951 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |