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 7191 | . . . . . 6 ⊢ (0[,]2) ∈ V | |
3 | 1, 2 | eqeltri 2911 | . . . . 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 42624 | . . 3 ⊢ (⊤ → 𝑆 ∈ SAlg) |
7 | 6 | mptru 1544 | . 2 ⊢ 𝑆 ∈ SAlg |
8 | salexct3.x | . . 3 ⊢ 𝑋 = ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) | |
9 | 0re 10645 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℝ | |
10 | 2re 11714 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ | |
11 | 9, 10 | pm3.2i 473 | . . . . . . . . . . 11 ⊢ (0 ∈ ℝ ∧ 2 ∈ ℝ) |
12 | 9 | leidi 11176 | . . . . . . . . . . . 12 ⊢ 0 ≤ 0 |
13 | 1le2 11849 | . . . . . . . . . . . 12 ⊢ 1 ≤ 2 | |
14 | 12, 13 | pm3.2i 473 | . . . . . . . . . . 11 ⊢ (0 ≤ 0 ∧ 1 ≤ 2) |
15 | iccss 12807 | . . . . . . . . . . 11 ⊢ (((0 ∈ ℝ ∧ 2 ∈ ℝ) ∧ (0 ≤ 0 ∧ 1 ≤ 2)) → (0[,]1) ⊆ (0[,]2)) | |
16 | 11, 14, 15 | mp2an 690 | . . . . . . . . . 10 ⊢ (0[,]1) ⊆ (0[,]2) |
17 | id 22 | . . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ (0[,]1)) | |
18 | 16, 17 | sseldi 3967 | . . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ (0[,]2)) |
19 | 18, 1 | eleqtrrdi 2926 | . . . . . . . 8 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ 𝐴) |
20 | snelpwi 5339 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ∈ 𝒫 𝐴) | |
21 | 19, 20 | syl 17 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]1) → {𝑦} ∈ 𝒫 𝐴) |
22 | snfi 8596 | . . . . . . . . . 10 ⊢ {𝑦} ∈ Fin | |
23 | fict 9118 | . . . . . . . . . 10 ⊢ ({𝑦} ∈ Fin → {𝑦} ≼ ω) | |
24 | 22, 23 | ax-mp 5 | . . . . . . . . 9 ⊢ {𝑦} ≼ ω |
25 | orc 863 | . . . . . . . . 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 514 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]1) → ({𝑦} ∈ 𝒫 𝐴 ∧ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
29 | breq1 5071 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → (𝑥 ≼ ω ↔ {𝑦} ≼ ω)) | |
30 | difeq2 4095 | . . . . . . . . 9 ⊢ (𝑥 = {𝑦} → (𝐴 ∖ 𝑥) = (𝐴 ∖ {𝑦})) | |
31 | 30 | breq1d 5078 | . . . . . . . 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 3685 | . . . . . 6 ⊢ ({𝑦} ∈ 𝑆 ↔ ({𝑦} ∈ 𝒫 𝐴 ∧ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
34 | 28, 33 | sylibr 236 | . . . . 5 ⊢ (𝑦 ∈ (0[,]1) → {𝑦} ∈ 𝑆) |
35 | 34 | rgen 3150 | . . . 4 ⊢ ∀𝑦 ∈ (0[,]1){𝑦} ∈ 𝑆 |
36 | eqid 2823 | . . . . 5 ⊢ (𝑦 ∈ (0[,]1) ↦ {𝑦}) = (𝑦 ∈ (0[,]1) ↦ {𝑦}) | |
37 | 36 | rnmptss 6888 | . . . 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 4003 | . 2 ⊢ 𝑋 ⊆ 𝑆 |
40 | 8 | unieqi 4853 | . . . . 5 ⊢ ∪ 𝑋 = ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) |
41 | snex 5334 | . . . . . . . 8 ⊢ {𝑦} ∈ V | |
42 | 41 | rgenw 3152 | . . . . . . 7 ⊢ ∀𝑦 ∈ (0[,]1){𝑦} ∈ V |
43 | dfiun3g 5837 | . . . . . . 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 2832 | . . . . 5 ⊢ ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) = ∪ 𝑦 ∈ (0[,]1){𝑦} |
46 | iunid 4986 | . . . . 5 ⊢ ∪ 𝑦 ∈ (0[,]1){𝑦} = (0[,]1) | |
47 | 40, 45, 46 | 3eqtrri 2851 | . . . 4 ⊢ (0[,]1) = ∪ 𝑋 |
48 | 47 | eqcomi 2832 | . . 3 ⊢ ∪ 𝑋 = (0[,]1) |
49 | 1, 5, 48 | salexct2 42629 | . 2 ⊢ ¬ ∪ 𝑋 ∈ 𝑆 |
50 | 7, 39, 49 | 3pm3.2i 1335 | 1 ⊢ (𝑆 ∈ SAlg ∧ 𝑋 ⊆ 𝑆 ∧ ¬ ∪ 𝑋 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ∀wral 3140 {crab 3144 Vcvv 3496 ∖ cdif 3935 ⊆ wss 3938 𝒫 cpw 4541 {csn 4569 ∪ cuni 4840 ∪ ciun 4921 class class class wbr 5068 ↦ cmpt 5148 ran crn 5558 (class class class)co 7158 ωcom 7582 ≼ cdom 8509 Fincfn 8511 ℝcr 10538 0cc0 10539 1c1 10540 ≤ cle 10678 2c2 11695 [,]cicc 12744 SAlgcsalg 42600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cc 9859 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-top 21504 df-topon 21521 df-bases 21556 df-ntr 21630 df-salg 42601 |
This theorem is referenced by: (None) |
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