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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 11263 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℝ | |
| 3 | 2re 12340 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ | |
| 4 | 2, 3 | pm3.2i 470 | . . . . . . . . . . 11 ⊢ (0 ∈ ℝ ∧ 2 ∈ ℝ) |
| 5 | 2 | leidi 11797 | . . . . . . . . . . . 12 ⊢ 0 ≤ 0 |
| 6 | 1le2 12475 | . . . . . . . . . . . 12 ⊢ 1 ≤ 2 | |
| 7 | 5, 6 | pm3.2i 470 | . . . . . . . . . . 11 ⊢ (0 ≤ 0 ∧ 1 ≤ 2) |
| 8 | iccss 13455 | . . . . . . . . . . 11 ⊢ (((0 ∈ ℝ ∧ 2 ∈ ℝ) ∧ (0 ≤ 0 ∧ 1 ≤ 2)) → (0[,]1) ⊆ (0[,]2)) | |
| 9 | 4, 7, 8 | mp2an 692 | . . . . . . . . . 10 ⊢ (0[,]1) ⊆ (0[,]2) |
| 10 | id 22 | . . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ (0[,]1)) | |
| 11 | 9, 10 | sselid 3981 | . . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ (0[,]2)) |
| 12 | salgensscntex.a | . . . . . . . . 9 ⊢ 𝐴 = (0[,]2) | |
| 13 | 11, 12 | eleqtrrdi 2852 | . . . . . . . 8 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ 𝐴) |
| 14 | snelpwi 5448 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ∈ 𝒫 𝐴) | |
| 15 | 13, 14 | syl 17 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]1) → {𝑦} ∈ 𝒫 𝐴) |
| 16 | snfi 9083 | . . . . . . . . . 10 ⊢ {𝑦} ∈ Fin | |
| 17 | fict 9693 | . . . . . . . . . 10 ⊢ ({𝑦} ∈ Fin → {𝑦} ≼ ω) | |
| 18 | 16, 17 | ax-mp 5 | . . . . . . . . 9 ⊢ {𝑦} ≼ ω |
| 19 | orc 868 | . . . . . . . . 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 511 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]1) → ({𝑦} ∈ 𝒫 𝐴 ∧ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
| 23 | breq1 5146 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → (𝑥 ≼ ω ↔ {𝑦} ≼ ω)) | |
| 24 | difeq2 4120 | . . . . . . . . 9 ⊢ (𝑥 = {𝑦} → (𝐴 ∖ 𝑥) = (𝐴 ∖ {𝑦})) | |
| 25 | 24 | breq1d 5153 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ {𝑦}) ≼ ω)) |
| 26 | 23, 25 | orbi12d 919 | . . . . . . 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 3695 | . . . . . 6 ⊢ ({𝑦} ∈ 𝑆 ↔ ({𝑦} ∈ 𝒫 𝐴 ∧ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
| 29 | 22, 28 | sylibr 234 | . . . . 5 ⊢ (𝑦 ∈ (0[,]1) → {𝑦} ∈ 𝑆) |
| 30 | 29 | rgen 3063 | . . . 4 ⊢ ∀𝑦 ∈ (0[,]1){𝑦} ∈ 𝑆 |
| 31 | eqid 2737 | . . . . 5 ⊢ (𝑦 ∈ (0[,]1) ↦ {𝑦}) = (𝑦 ∈ (0[,]1) ↦ {𝑦}) | |
| 32 | 31 | rnmptss 7143 | . . . 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 4030 | . 2 ⊢ 𝑋 ⊆ 𝑆 |
| 35 | ovex 7464 | . . . . . 6 ⊢ (0[,]2) ∈ V | |
| 36 | 12, 35 | eqeltri 2837 | . . . . 5 ⊢ 𝐴 ∈ V |
| 37 | 36 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐴 ∈ V) |
| 38 | 37, 27 | salexct 46349 | . . 3 ⊢ (⊤ → 𝑆 ∈ SAlg) |
| 39 | 38 | mptru 1547 | . 2 ⊢ 𝑆 ∈ SAlg |
| 40 | ovex 7464 | . . . . . . . . 9 ⊢ (0[,]1) ∈ V | |
| 41 | 40 | mptex 7243 | . . . . . . . 8 ⊢ (𝑦 ∈ (0[,]1) ↦ {𝑦}) ∈ V |
| 42 | 41 | rnex 7932 | . . . . . . 7 ⊢ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) ∈ V |
| 43 | 1, 42 | eqeltri 2837 | . . . . . 6 ⊢ 𝑋 ∈ V |
| 44 | 43 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝑋 ∈ V) |
| 45 | salgensscntex.g | . . . . 5 ⊢ 𝐺 = (SalGen‘𝑋) | |
| 46 | 1 | unieqi 4919 | . . . . . 6 ⊢ ∪ 𝑋 = ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) |
| 47 | vsnex 5434 | . . . . . . . . 9 ⊢ {𝑦} ∈ V | |
| 48 | 47 | rgenw 3065 | . . . . . . . 8 ⊢ ∀𝑦 ∈ (0[,]1){𝑦} ∈ V |
| 49 | dfiun3g 5978 | . . . . . . . 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 2746 | . . . . . 6 ⊢ ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) = ∪ 𝑦 ∈ (0[,]1){𝑦} |
| 52 | iunid 5060 | . . . . . 6 ⊢ ∪ 𝑦 ∈ (0[,]1){𝑦} = (0[,]1) | |
| 53 | 46, 51, 52 | 3eqtrri 2770 | . . . . 5 ⊢ (0[,]1) = ∪ 𝑋 |
| 54 | 44, 45, 53 | unisalgen 46355 | . . . 4 ⊢ (⊤ → (0[,]1) ∈ 𝐺) |
| 55 | 54 | mptru 1547 | . . 3 ⊢ (0[,]1) ∈ 𝐺 |
| 56 | eqid 2737 | . . . 4 ⊢ (0[,]1) = (0[,]1) | |
| 57 | 12, 27, 56 | salexct2 46354 | . . 3 ⊢ ¬ (0[,]1) ∈ 𝑆 |
| 58 | nelss 4049 | . . 3 ⊢ (((0[,]1) ∈ 𝐺 ∧ ¬ (0[,]1) ∈ 𝑆) → ¬ 𝐺 ⊆ 𝑆) | |
| 59 | 55, 57, 58 | mp2an 692 | . 2 ⊢ ¬ 𝐺 ⊆ 𝑆 |
| 60 | 34, 39, 59 | 3pm3.2i 1340 | 1 ⊢ (𝑋 ⊆ 𝑆 ∧ 𝑆 ∈ SAlg ∧ ¬ 𝐺 ⊆ 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 ∀wral 3061 {crab 3436 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 𝒫 cpw 4600 {csn 4626 ∪ cuni 4907 ∪ ciun 4991 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ωcom 7887 ≼ cdom 8983 Fincfn 8985 ℝcr 11154 0cc0 11155 1c1 11156 ≤ cle 11296 2c2 12321 [,]cicc 13390 SAlgcsalg 46323 SalGencsalgen 46327 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 df-ntr 23028 df-salg 46324 df-salgen 46328 |
| This theorem is referenced by: (None) |
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